Complex number calculator

There are 24 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains roots of complex numbers or powers to 1/n.

z₁ = ((-5i)^(1/8))*(8^(1/3)) = 2.3986959-0.4771303i = 2.4456891 × e^{i -0.1963495} = 2.4456891 × e^{i (-0.0625) π} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i -0.1963495} = 1.2228445 × e^{i (-0.0625) π} = 1.1993479-0.2385651i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (1.1993479-0.2385651i) * 2 = 2.3986959-0.4771303i

The result z₁

Rectangular form (standard form):
z = 2.3986959-0.4771303i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -11°15'

Polar form:
z = 2.4456891 × (cos (-11°15') + i sin (-11°15'))

Exponential form:
z = 2.4456891 × e^{i -0.1963495} = 2.4456891 × e^{i (-0.0625) π}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.1963495 rad = -11.25° = -11°15' = -0.0625π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.3986959-0.4771303i
Real part: x = Re z = 2.399
Imaginary part: y = Im z = -0.47713027

z₂ = ((-5i)^(1/8))*(8^(1/3)) = 2.0335162+1.3587521i = 2.4456891 × e^{i 3π/16} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 3π/16} = 1.0167581+0.679376i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (1.0167581+0.679376i) * 2 = 2.0335162+1.3587521i

The result z₂

Rectangular form (standard form):
z = 2.0335162+1.3587521i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 33°45'

Polar form:
z = 2.4456891 × (cos 33°45' + i sin 33°45')

Exponential form:
z = 2.4456891 × e^{i 0.5890486} = 2.4456891 × e^{i 3π/16}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.5890486 rad = 33.75° = 33°45' = 0.1875π = 3π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.0335162+1.3587521i
Real part: x = Re z = 2.034
Imaginary part: y = Im z = 1.35875206

z₃ = ((-5i)^(1/8))*(8^(1/3)) = 0.4771303+2.3986959i = 2.4456891 × e^{i 7π/16} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 7π/16} = 0.2385651+1.1993479i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (0.2385651+1.1993479i) * 2 = 0.4771303+2.3986959i

The result z₃

Rectangular form (standard form):
z = 0.4771303+2.3986959i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 78°45'

Polar form:
z = 2.4456891 × (cos 78°45' + i sin 78°45')

Exponential form:
z = 2.4456891 × e^{i 1.3744468} = 2.4456891 × e^{i 7π/16}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.3744468 rad = 78.75° = 78°45' = 0.4375π = 7π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.4771303+2.3986959i
Real part: x = Re z = 0.477
Imaginary part: y = Im z = 2.39869586

z₄ = ((-5i)^(1/8))*(8^(1/3)) = -1.3587521+2.0335162i = 2.4456891 × e^{i 11π/16} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 11π/16} = -0.679376+1.0167581i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (-0.679376+1.0167581i) * 2 = -1.3587521+2.0335162i

The result z₄

Rectangular form (standard form):
z = -1.3587521+2.0335162i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 123°45'

Polar form:
z = 2.4456891 × (cos 123°45' + i sin 123°45')

Exponential form:
z = 2.4456891 × e^{i 2.1598449} = 2.4456891 × e^{i 11π/16}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.1598449 rad = 123.75° = 123°45' = 0.6875π = 11π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.3587521+2.0335162i
Real part: x = Re z = -1.359
Imaginary part: y = Im z = 2.03351616

z₅ = ((-5i)^(1/8))*(8^(1/3)) = -2.3986959+0.4771303i = 2.4456891 × e^{i 15π/16} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 15π/16} = -1.1993479+0.2385651i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (-1.1993479+0.2385651i) * 2 = -2.3986959+0.4771303i

The result z₅

Rectangular form (standard form):
z = -2.3986959+0.4771303i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 168°45'

Polar form:
z = 2.4456891 × (cos 168°45' + i sin 168°45')

Exponential form:
z = 2.4456891 × e^{i 2.9452431} = 2.4456891 × e^{i 15π/16}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.9452431 rad = 168.75° = 168°45' = 0.9375π = 15π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.3986959+0.4771303i
Real part: x = Re z = -2.399
Imaginary part: y = Im z = 0.47713027

z₆ = ((-5i)^(1/8))*(8^(1/3)) = -2.0335162-1.3587521i = 2.4456891 × e^{i (-13π/16)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-13π/16)} = -1.0167581-0.679376i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (-1.0167581-0.679376i) * 2 = -2.0335162-1.3587521i

The result z₆

Rectangular form (standard form):
z = -2.0335162-1.3587521i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -146°15'

Polar form:
z = 2.4456891 × (cos (-146°15') + i sin (-146°15'))

Exponential form:
z = 2.4456891 × e^{i -2.552544} = 2.4456891 × e^{i (-13π/16)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.552544 rad = -146.25° = -146°15' = -0.8125π = -13π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.0335162-1.3587521i
Real part: x = Re z = -2.034
Imaginary part: y = Im z = -1.35875206

z₇ = ((-5i)^(1/8))*(8^(1/3)) = -0.4771303-2.3986959i = 2.4456891 × e^{i (-9π/16)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-9π/16)} = -0.2385651-1.1993479i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (-0.2385651-1.1993479i) * 2 = -0.4771303-2.3986959i

The result z₇

Rectangular form (standard form):
z = -0.4771303-2.3986959i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -101°15'

Polar form:
z = 2.4456891 × (cos (-101°15') + i sin (-101°15'))

Exponential form:
z = 2.4456891 × e^{i -1.7671459} = 2.4456891 × e^{i (-9π/16)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -1.7671459 rad = -101.25° = -101°15' = -0.5625π = -9π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.4771303-2.3986959i
Real part: x = Re z = -0.477
Imaginary part: y = Im z = -2.39869586

z₈ = ((-5i)^(1/8))*(8^(1/3)) = 1.3587521-2.0335162i = 2.4456891 × e^{i (-5π/16)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-5π/16)} = 0.679376-1.0167581i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = 2
Multiple: the result of step No. 2 * the result of step No. 4 = (0.679376-1.0167581i) * 2 = 1.3587521-2.0335162i

The result z₈

Rectangular form (standard form):
z = 1.3587521-2.0335162i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -56°15'

Polar form:
z = 2.4456891 × (cos (-56°15') + i sin (-56°15'))

Exponential form:
z = 2.4456891 × e^{i -0.9817477} = 2.4456891 × e^{i (-5π/16)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.9817477 rad = -56.25° = -56°15' = -0.3125π = -5π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.3587521-2.0335162i
Real part: x = Re z = 1.359
Imaginary part: y = Im z = -2.03351616

z₉ = ((-5i)^(1/8))*(8^(1/3)) = -0.786141+2.3158967i = 2.4456891 × e^{i 29π/48} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i -0.1963495} = 1.2228445 × e^{i (-0.0625) π} = 1.1993479-0.2385651i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (1.1993479-0.2385651i) * (-1+1.7320508i) = 1.19934793 * (-1) + 1.19934793 * 1.7320508076i + (-0.2385651361i) * (-1) + (-0.2385651361i) * 1.7320508076i = -1.19934793+2.07733155i+0.23856514i-0.41320694i² = -1.19934793+2.07733155i+0.23856514i+0.41320694 = -1.19934793 0.413207 +i(2.07733155 + 0.238565) = -0.786141+2.3158967i
alternative steps

1.2228445 × e^{i -0.1963495} = 1.2228445 × e^{i (-0.0625) π} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i ((-0.0625)+2π/3)} = 2.4456891 × e^{i 29π/48} = -0.786141+2.3158967i

The result z₉

Rectangular form (standard form):
z = -0.786141+2.3158967i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 108°45'

Polar form:
z = 2.4456891 × (cos 108°45' + i sin 108°45')

Exponential form:
z = 2.4456891 × e^{i 1.8980456} = 2.4456891 × e^{i 29π/48}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.8980456 rad = 108.75° = 108°45' = 0.6041667π = 29π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.786141+2.3158967i
Real part: x = Re z = -0.786
Imaginary part: y = Im z = 2.31589669

z₁₀ = ((-5i)^(1/8))*(8^(1/3)) = -2.1934719+1.0817006i = 2.4456891 × e^{i 41π/48} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 3π/16} = 1.0167581+0.679376i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (1.0167581+0.679376i) * (-1+1.7320508i) = 1.0167580798154 * (-1) + 1.0167580798154 * 1.7320508076i + 0.67937602882697i * (-1) + 0.67937602882697i * 1.7320508076i = -1.01675808+1.76107665i-0.67937603i+1.1767138i² = -1.01675808+1.76107665i-0.67937603i-1.1767138 = -1.01675808- 1.176714 +i(1.76107665 - 0.679376) = -2.1934719+1.0817006i
alternative steps

1.2228445 × e^{i 3π/16} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i (3π/16+2π/3)} = 2.4456891 × e^{i 41π/48} = -2.1934719+1.0817006i

The result z₁₀

Rectangular form (standard form):
z = -2.1934719+1.0817006i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 153°45'

Polar form:
z = 2.4456891 × (cos 153°45' + i sin 153°45')

Exponential form:
z = 2.4456891 × e^{i 2.6834437} = 2.4456891 × e^{i 41π/48}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.6834437 rad = 153.75° = 153°45' = 0.8541667π = 41π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.1934719+1.0817006i
Real part: x = Re z = -2.193
Imaginary part: y = Im z = 1.08170062

z₁₁ = ((-5i)^(1/8))*(8^(1/3)) = -2.3158967-0.786141i = 2.4456891 × e^{i (-43π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 7π/16} = 0.2385651+1.1993479i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (0.2385651+1.1993479i) * (-1+1.7320508i) = 0.2385651361 * (-1) + 0.2385651361 * 1.7320508076i + 1.19934793i * (-1) + 1.19934793i * 1.7320508076i = -0.23856514+0.41320694i-1.19934793i+2.07733155i² = -0.23856514+0.41320694i-1.19934793i-2.07733155 = -0.23856514- 2.077332 +i(0.41320694 - 1.199348) = -2.3158967-0.786141i
alternative steps

1.2228445 × e^{i 7π/16} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i (7π/16+2π/3)} = 2.4456891 × e^{i (-43π/48)} = -2.3158967-0.786141i

The result z₁₁

Rectangular form (standard form):
z = -2.3158967-0.786141i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -161°15'

Polar form:
z = 2.4456891 × (cos (-161°15') + i sin (-161°15'))

Exponential form:
z = 2.4456891 × e^{i -2.8143434} = 2.4456891 × e^{i (-43π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.8143434 rad = -161.25° = -161°15' = -0.8958333π = -43π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.3158967-0.786141i
Real part: x = Re z = -2.316
Imaginary part: y = Im z = -0.78614099

z₁₂ = ((-5i)^(1/8))*(8^(1/3)) = -1.0817006-2.1934719i = 2.4456891 × e^{i (-31π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 11π/16} = -0.679376+1.0167581i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-0.679376+1.0167581i) * (-1+1.7320508i) = -0.67937602882697 * (-1) + (-0.67937602882697) * 1.7320508076i + 1.0167580798154i * (-1) + 1.0167580798154i * 1.7320508076i = 0.67937603-1.1767138i-1.01675808i+1.76107665i² = 0.67937603-1.1767138i-1.01675808i-1.76107665 = 0.67937603- 1.761077 +i(-1.1767138 - 1.016758) = -1.0817006-2.1934719i
alternative steps

1.2228445 × e^{i 11π/16} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i (11π/16+2π/3)} = 2.4456891 × e^{i (-31π/48)} = -1.0817006-2.1934719i

The result z₁₂

Rectangular form (standard form):
z = -1.0817006-2.1934719i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -116°15'

Polar form:
z = 2.4456891 × (cos (-116°15') + i sin (-116°15'))

Exponential form:
z = 2.4456891 × e^{i -2.0289453} = 2.4456891 × e^{i (-31π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.0289453 rad = -116.25° = -116°15' = -0.6458333π = -31π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.0817006-2.1934719i
Real part: x = Re z = -1.082
Imaginary part: y = Im z = -2.19347188

z₁₃ = ((-5i)^(1/8))*(8^(1/3)) = 0.786141-2.3158967i = 2.4456891 × e^{i (-19π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 15π/16} = -1.1993479+0.2385651i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-1.1993479+0.2385651i) * (-1+1.7320508i) = -1.19934793 * (-1) + (-1.19934793) * 1.7320508076i + 0.2385651361i * (-1) + 0.2385651361i * 1.7320508076i = 1.19934793-2.07733155i-0.23856514i+0.41320694i² = 1.19934793-2.07733155i-0.23856514i-0.41320694 = 1.19934793- 0.413207 +i(-2.07733155 - 0.238565) = 0.786141-2.3158967i
alternative steps

1.2228445 × e^{i 15π/16} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i (15π/16+2π/3)} = 2.4456891 × e^{i (-19π/48)} = 0.786141-2.3158967i

The result z₁₃

Rectangular form (standard form):
z = 0.786141-2.3158967i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -71°15'

Polar form:
z = 2.4456891 × (cos (-71°15') + i sin (-71°15'))

Exponential form:
z = 2.4456891 × e^{i -1.2435471} = 2.4456891 × e^{i (-19π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -1.2435471 rad = -71.25° = -71°15' = -0.3958333π = -19π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.786141-2.3158967i
Real part: x = Re z = 0.786
Imaginary part: y = Im z = -2.31589669

z₁₄ = ((-5i)^(1/8))*(8^(1/3)) = 2.1934719-1.0817006i = 2.4456891 × e^{i (-7π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-13π/16)} = -1.0167581-0.679376i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-1.0167581-0.679376i) * (-1+1.7320508i) = -1.0167580798154 * (-1) + (-1.0167580798154) * 1.7320508076i + (-0.67937602882697i) * (-1) + (-0.67937602882697i) * 1.7320508076i = 1.01675808-1.76107665i+0.67937603i-1.1767138i² = 1.01675808-1.76107665i+0.67937603i+1.1767138 = 1.01675808 1.176714 +i(-1.76107665 + 0.679376) = 2.1934719-1.0817006i
alternative steps

1.2228445 × e^{i (-13π/16)} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i ((-13π/16)+2π/3)} = 2.4456891 × e^{i (-7π/48)} = 2.1934719-1.0817006i

The result z₁₄

Rectangular form (standard form):
z = 2.1934719-1.0817006i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -26°15'

Polar form:
z = 2.4456891 × (cos (-26°15') + i sin (-26°15'))

Exponential form:
z = 2.4456891 × e^{i -0.4581489} = 2.4456891 × e^{i (-7π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.4581489 rad = -26.25° = -26°15' = -0.1458333π = -7π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.1934719-1.0817006i
Real part: x = Re z = 2.193
Imaginary part: y = Im z = -1.08170062

z₁₅ = ((-5i)^(1/8))*(8^(1/3)) = 2.3158967+0.786141i = 2.4456891 × e^{i 0.3272492} = 2.4456891 × e^{i 0.1041667 π} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-9π/16)} = -0.2385651-1.1993479i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-0.2385651-1.1993479i) * (-1+1.7320508i) = -0.2385651361 * (-1) + (-0.2385651361) * 1.7320508076i + (-1.19934793i) * (-1) + (-1.19934793i) * 1.7320508076i = 0.23856514-0.41320694i+1.19934793i-2.07733155i² = 0.23856514-0.41320694i+1.19934793i+2.07733155 = 0.23856514 2.077332 +i(-0.41320694 + 1.199348) = 2.3158967+0.786141i
alternative steps

1.2228445 × e^{i (-9π/16)} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i ((-9π/16)+2π/3)} = 2.4456891 × e^{i 0.3272492} = 2.4456891 × e^{i 0.1041667 π} = 2.3158967+0.786141i

The result z₁₅

Rectangular form (standard form):
z = 2.3158967+0.786141i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 18°45'

Polar form:
z = 2.4456891 × (cos 18°45' + i sin 18°45')

Exponential form:
z = 2.4456891 × e^{i 0.3272492} = 2.4456891 × e^{i 0.1041667 π}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.3272492 rad = 18.75° = 18°45' = 0.1041667π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.3158967+0.786141i
Real part: x = Re z = 2.316
Imaginary part: y = Im z = 0.78614099

z₁₆ = ((-5i)^(1/8))*(8^(1/3)) = 1.0817006+2.1934719i = 2.4456891 × e^{i 17π/48} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-5π/16)} = 0.679376-1.0167581i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1+1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (0.679376-1.0167581i) * (-1+1.7320508i) = 0.67937602882697 * (-1) + 0.67937602882697 * 1.7320508076i + (-1.0167580798154i) * (-1) + (-1.0167580798154i) * 1.7320508076i = -0.67937603+1.1767138i+1.01675808i-1.76107665i² = -0.67937603+1.1767138i+1.01675808i+1.76107665 = -0.67937603 1.761077 +i(1.1767138 + 1.016758) = 1.0817006+2.1934719i
alternative steps

1.2228445 × e^{i (-5π/16)} × 2 × e^{i 2π/3} = 1.2228445 × 2 × e^{i ((-5π/16)+2π/3)} = 2.4456891 × e^{i 17π/48} = 1.0817006+2.1934719i

The result z₁₆

Rectangular form (standard form):
z = 1.0817006+2.1934719i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 63°45'

Polar form:
z = 2.4456891 × (cos 63°45' + i sin 63°45')

Exponential form:
z = 2.4456891 × e^{i 1.1126474} = 2.4456891 × e^{i 17π/48}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.1126474 rad = 63.75° = 63°45' = 0.3541667π = 17π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.0817006+2.1934719i
Real part: x = Re z = 1.082
Imaginary part: y = Im z = 2.19347188

z₁₇ = ((-5i)^(1/8))*(8^(1/3)) = -1.6125549-1.8387664i = 2.4456891 × e^{i (-35π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i -0.1963495} = 1.2228445 × e^{i (-0.0625) π} = 1.1993479-0.2385651i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (1.1993479-0.2385651i) * (-1-1.7320508i) = 1.19934793 * (-1) + 1.19934793 * (-1.7320508076i) + (-0.2385651361i) * (-1) + (-0.2385651361i) * (-1.7320508076i) = -1.19934793-2.07733155i+0.23856514i+0.41320694i² = -1.19934793-2.07733155i+0.23856514i-0.41320694 = -1.19934793- 0.413207 +i(-2.07733155 + 0.238565) = -1.6125549-1.8387664i
alternative steps

1.2228445 × e^{i -0.1963495} = 1.2228445 × e^{i (-0.0625) π} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i ((-0.0625)+(-2π/3))} = 2.4456891 × e^{i (-35π/48)} = -1.6125549-1.8387664i

The result z₁₇

Rectangular form (standard form):
z = -1.6125549-1.8387664i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -131°15'

Polar form:
z = 2.4456891 × (cos (-131°15') + i sin (-131°15'))

Exponential form:
z = 2.4456891 × e^{i -2.2907446} = 2.4456891 × e^{i (-35π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.2907446 rad = -131.25° = -131°15' = -0.7291667π = -35π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.6125549-1.8387664i
Real part: x = Re z = -1.613
Imaginary part: y = Im z = -1.83876641

z₁₈ = ((-5i)^(1/8))*(8^(1/3)) = 0.1599557-2.4404527i = 2.4456891 × e^{i (-23π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 3π/16} = 1.0167581+0.679376i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (1.0167581+0.679376i) * (-1-1.7320508i) = 1.0167580798154 * (-1) + 1.0167580798154 * (-1.7320508076i) + 0.67937602882697i * (-1) + 0.67937602882697i * (-1.7320508076i) = -1.01675808-1.76107665i-0.67937603i-1.1767138i² = -1.01675808-1.76107665i-0.67937603i+1.1767138 = -1.01675808 1.176714 +i(-1.76107665 - 0.679376) = 0.1599557-2.4404527i
alternative steps

1.2228445 × e^{i 3π/16} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i (3π/16+(-2π/3))} = 2.4456891 × e^{i (-23π/48)} = 0.1599557-2.4404527i

The result z₁₈

Rectangular form (standard form):
z = 0.1599557-2.4404527i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -86°15'

Polar form:
z = 2.4456891 × (cos (-86°15') + i sin (-86°15'))

Exponential form:
z = 2.4456891 × e^{i -1.5053465} = 2.4456891 × e^{i (-23π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -1.5053465 rad = -86.25° = -86°15' = -0.4791667π = -23π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.1599557-2.4404527i
Real part: x = Re z = 0.16
Imaginary part: y = Im z = -2.44045268

z₁₉ = ((-5i)^(1/8))*(8^(1/3)) = 1.8387664-1.6125549i = 2.4456891 × e^{i (-11π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 7π/16} = 0.2385651+1.1993479i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (0.2385651+1.1993479i) * (-1-1.7320508i) = 0.2385651361 * (-1) + 0.2385651361 * (-1.7320508076i) + 1.19934793i * (-1) + 1.19934793i * (-1.7320508076i) = -0.23856514-0.41320694i-1.19934793i-2.07733155i² = -0.23856514-0.41320694i-1.19934793i+2.07733155 = -0.23856514 2.077332 +i(-0.41320694 - 1.199348) = 1.8387664-1.6125549i
alternative steps

1.2228445 × e^{i 7π/16} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i (7π/16+(-2π/3))} = 2.4456891 × e^{i (-11π/48)} = 1.8387664-1.6125549i

The result z₁₉

Rectangular form (standard form):
z = 1.8387664-1.6125549i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -41°15'

Polar form:
z = 2.4456891 × (cos (-41°15') + i sin (-41°15'))

Exponential form:
z = 2.4456891 × e^{i -0.7199483} = 2.4456891 × e^{i (-11π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.7199483 rad = -41.25° = -41°15' = -0.2291667π = -11π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.8387664-1.6125549i
Real part: x = Re z = 1.839
Imaginary part: y = Im z = -1.61255487

z₂₀ = ((-5i)^(1/8))*(8^(1/3)) = 2.4404527+0.1599557i = 2.4456891 × e^{i 0.0654498} = 2.4456891 × e^{i 0.0208333 π} Calculation steps principal root

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 11π/16} = -0.679376+1.0167581i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-0.679376+1.0167581i) * (-1-1.7320508i) = -0.67937602882697 * (-1) + (-0.67937602882697) * (-1.7320508076i) + 1.0167580798154i * (-1) + 1.0167580798154i * (-1.7320508076i) = 0.67937603+1.1767138i-1.01675808i-1.76107665i² = 0.67937603+1.1767138i-1.01675808i+1.76107665 = 0.67937603 1.761077 +i(1.1767138 - 1.016758) = 2.4404527+0.1599557i
alternative steps

1.2228445 × e^{i 11π/16} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i (11π/16+(-2π/3))} = 2.4456891 × e^{i 0.0654498} = 2.4456891 × e^{i 0.0208333 π} = 2.4404527+0.1599557i

The result z₂₀

Rectangular form (standard form):
z = 2.4404527+0.1599557i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 3°45'

Polar form:
z = 2.4456891 × (cos 3°45' + i sin 3°45')

Exponential form:
z = 2.4456891 × e^{i 0.0654498} = 2.4456891 × e^{i 0.0208333 π}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.0654498 rad = 3.75° = 3°45' = 0.0208333π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.4404527+0.1599557i
Real part: x = Re z = 2.44
Imaginary part: y = Im z = 0.15995572

z₂₁ = ((-5i)^(1/8))*(8^(1/3)) = 1.6125549+1.8387664i = 2.4456891 × e^{i 13π/48} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i 15π/16} = -1.1993479+0.2385651i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-1.1993479+0.2385651i) * (-1-1.7320508i) = -1.19934793 * (-1) + (-1.19934793) * (-1.7320508076i) + 0.2385651361i * (-1) + 0.2385651361i * (-1.7320508076i) = 1.19934793+2.07733155i-0.23856514i-0.41320694i² = 1.19934793+2.07733155i-0.23856514i+0.41320694 = 1.19934793 0.413207 +i(2.07733155 - 0.238565) = 1.6125549+1.8387664i
alternative steps

1.2228445 × e^{i 15π/16} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i (15π/16+(-2π/3))} = 2.4456891 × e^{i 13π/48} = 1.6125549+1.8387664i

The result z₂₁

Rectangular form (standard form):
z = 1.6125549+1.8387664i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 48°45'

Polar form:
z = 2.4456891 × (cos 48°45' + i sin 48°45')

Exponential form:
z = 2.4456891 × e^{i 0.850848} = 2.4456891 × e^{i 13π/48}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.850848 rad = 48.75° = 48°45' = 0.2708333π = 13π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.6125549+1.8387664i
Real part: x = Re z = 1.613
Imaginary part: y = Im z = 1.83876641

z₂₂ = ((-5i)^(1/8))*(8^(1/3)) = -0.1599557+2.4404527i = 2.4456891 × e^{i 25π/48} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-13π/16)} = -1.0167581-0.679376i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-1.0167581-0.679376i) * (-1-1.7320508i) = -1.0167580798154 * (-1) + (-1.0167580798154) * (-1.7320508076i) + (-0.67937602882697i) * (-1) + (-0.67937602882697i) * (-1.7320508076i) = 1.01675808+1.76107665i+0.67937603i+1.1767138i² = 1.01675808+1.76107665i+0.67937603i-1.1767138 = 1.01675808- 1.176714 +i(1.76107665 + 0.679376) = -0.1599557+2.4404527i
alternative steps

1.2228445 × e^{i (-13π/16)} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i ((-13π/16)+(-2π/3))} = 2.4456891 × e^{i 25π/48} = -0.1599557+2.4404527i

The result z₂₂

Rectangular form (standard form):
z = -0.1599557+2.4404527i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 93°45'

Polar form:
z = 2.4456891 × (cos 93°45' + i sin 93°45')

Exponential form:
z = 2.4456891 × e^{i 1.6362462} = 2.4456891 × e^{i 25π/48}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.6362462 rad = 93.75° = 93°45' = 0.5208333π = 25π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.1599557+2.4404527i
Real part: x = Re z = -0.16
Imaginary part: y = Im z = 2.44045268

z₂₃ = ((-5i)^(1/8))*(8^(1/3)) = -1.8387664+1.6125549i = 2.4456891 × e^{i 37π/48} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-9π/16)} = -0.2385651-1.1993479i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (-0.2385651-1.1993479i) * (-1-1.7320508i) = -0.2385651361 * (-1) + (-0.2385651361) * (-1.7320508076i) + (-1.19934793i) * (-1) + (-1.19934793i) * (-1.7320508076i) = 0.23856514+0.41320694i+1.19934793i+2.07733155i² = 0.23856514+0.41320694i+1.19934793i-2.07733155 = 0.23856514- 2.077332 +i(0.41320694 + 1.199348) = -1.8387664+1.6125549i
alternative steps

1.2228445 × e^{i (-9π/16)} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i ((-9π/16)+(-2π/3))} = 2.4456891 × e^{i 37π/48} = -1.8387664+1.6125549i

The result z₂₃

Rectangular form (standard form):
z = -1.8387664+1.6125549i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ 138°45'

Polar form:
z = 2.4456891 × (cos 138°45' + i sin 138°45')

Exponential form:
z = 2.4456891 × e^{i 2.4216443} = 2.4456891 × e^{i 37π/48}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.4216443 rad = 138.75° = 138°45' = 0.7708333π = 37π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.8387664+1.6125549i
Real part: x = Re z = -1.839
Imaginary part: y = Im z = 1.61255487

z₂₄ = ((-5i)^(1/8))*(8^(1/3)) = -2.4404527-0.1599557i = 2.4456891 × e^{i (-47π/48)} Calculation steps

Divide: 1 / 8 = 1/1 · 1/8 = 1 · 1/1 · 8 = 1/8 = 0.125
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 8/1 is 1/8) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by eight equals one eighth.
Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × e^{i (-π/2)})^0.125 = 5^0.125 × e^{i 0.125 × (-π/2)} = 1.2228445 × e^{i (-5π/16)} = 0.679376-1.0167581i
Divide: 1 / 3 = 1/1 · 1/3 = 1 · 1/1 · 3 = 1/3 = 0.33333333
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 3/1 is 1/3) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words, one divided by three equals one third.
Cube root: ∛8 = -1-1.7320508i
Multiple: the result of step No. 2 * the result of step No. 4 = (0.679376-1.0167581i) * (-1-1.7320508i) = 0.67937602882697 * (-1) + 0.67937602882697 * (-1.7320508076i) + (-1.0167580798154i) * (-1) + (-1.0167580798154i) * (-1.7320508076i) = -0.67937603-1.1767138i+1.01675808i+1.76107665i² = -0.67937603-1.1767138i+1.01675808i-1.76107665 = -0.67937603- 1.761077 +i(-1.1767138 + 1.016758) = -2.4404527-0.1599557i
alternative steps

1.2228445 × e^{i (-5π/16)} × 2 × e^{i (-2π/3)} = 1.2228445 × 2 × e^{i ((-5π/16)+(-2π/3))} = 2.4456891 × e^{i (-47π/48)} = -2.4404527-0.1599557i

The result z₂₄

Rectangular form (standard form):
z = -2.4404527-0.1599557i

Angle notation (phasor, module and argument):
z = 2.4456891 ∠ -176°15'

Polar form:
z = 2.4456891 × (cos (-176°15') + i sin (-176°15'))

Exponential form:
z = 2.4456891 × e^{i -3.0761428} = 2.4456891 × e^{i (-47π/48)}

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -3.0761428 rad = -176.25° = -176°15' = -0.9791667π = -47π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.4404527-0.1599557i
Real part: x = Re z = -2.44
Imaginary part: y = Im z = -0.15995572

This calculator performs all operations on complex numbers and evaluates expressions within the set of complex numbers. As an imaginary unit, use i or j (in electrical engineering), which satisfies the basic equation i² = −1 or j² = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

Basic operations with complex numbers

We hope that working with the complex number is quite easy because you can work with imaginary unit i as a variable. And use the definition i² = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

Addition

It is very simple: add up the real parts (without i) and add up the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i

(1+i) + (6-5i) = 7-4i
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5

Subtraction

Again it is very simple: subtract the real parts and subtract the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i

(1+i) - (3-5i) = -2+6i
-1/2 - (6-5i) = -6.5+5i
(10-5i) - (-5+5i) = 15-10i

Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and apply i² = -1.
This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i

(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i
-1/2 * (6-5i) = -3+2.5i
(10-5i) * (-5+5i) = -25+75i

Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the denominator's complex conjugate. This approach avoids imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it real), multiply with conjugate c-di:

(c+di)(c-di) = c²+d²

\frac{a + b i}{c + d i} = \frac{( a + b i ) ( c - d i )}{( c + d i ) ( c - d i )} = \frac{a c + b d + i ( b c - a d )}{c ^{2} + d ^{2}} = \frac{a c + b d}{c ^{2} + d ^{2}} + \frac{b c - a d}{c ^{2} + d ^{2}} i

(10-5i) / (1+i) = 2.5-7.5i
-3 / (2-i) = -1.2-0.6i
6i / (4+3i) = 0.72+0.96i

Absolute value or modulus

The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142136
|6i| = 6
abs(2+5i) = 5.3851648

Square root

The square root of a complex number (a+bi) is z, if z² = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is to use De Moivre's formula. Our calculator is on edge because the square root is not a well-defined function on a complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.1213203+2.1213203i
sqrt(10-6i) = 3.2910412-0.9115656i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225i
pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303i

Square, power, complex exponentiation

Our calculator can power any complex number to an integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:

i^{i} = e^{- π / 2}

i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224i
i^i = 0.2078795764
pow(1+i,3) = -2+2i

Functions

sqrt: Square Root of a value or expression.
sin: the sine of a value or expression. Autodetect radians/degrees.
cos: the cosine of a value or expression. Autodetect radians/degrees.
tan: tangent of a value or expression. Autodetect radians/degrees.
exp: e (the Euler Constant) raised to the power of a value or expression
pow: Power one complex number to another integer/real/complex number
ln: The natural logarithm of a value or expression
log: The base-10 logarithm of a value or expression
abs or |1+i|: The absolute value of a value or expression
phase: Phase (angle) of a complex number
cis: is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i
conj: the conjugate of a complex number - example: conj(4i+5) = 5-4i

Examples:

• cube root: cuberoot(1 - 27i)
• roots of Complex Numbers: pow(1 + i,1/7)
• phase, complex number angle: phase(1 + i)
• cis form complex numbers: 5 * cis(45°)
• The polar form of complex numbers: 10L60
• complex conjugate calculator: conj(4 + 5i)
• equation with complex numbers: (z + i/2 )/(1 - i) = 4z + 5i
• system of equations with imaginary numbers: x - y = 4 + 6i; 3ix + 7y=x + iy
• De Moivre's theorem - equation: z ^ 4=1
• multiplication of three complex numbers: (1 + 3i)(3 + 4i)(−5 + 3i)
• Find the product of 3-4i and its conjugate.: (3 - 4i) * conj(3 - 4i)
• operations with complex numbers: (3 - i) ^ 3

Complex number calculator

z1 = ((-5i)^(1/8))*(8^(1/3)) = 2.3986959-0.4771303i = 2.4456891 × ei -0.1963495 = 2.4456891 × ei (-0.0625) π Calculation steps

z2 = ((-5i)^(1/8))*(8^(1/3)) = 2.0335162+1.3587521i = 2.4456891 × ei 3π/16 Calculation steps

z3 = ((-5i)^(1/8))*(8^(1/3)) = 0.4771303+2.3986959i = 2.4456891 × ei 7π/16 Calculation steps

z4 = ((-5i)^(1/8))*(8^(1/3)) = -1.3587521+2.0335162i = 2.4456891 × ei 11π/16 Calculation steps

z5 = ((-5i)^(1/8))*(8^(1/3)) = -2.3986959+0.4771303i = 2.4456891 × ei 15π/16 Calculation steps

z6 = ((-5i)^(1/8))*(8^(1/3)) = -2.0335162-1.3587521i = 2.4456891 × ei (-13π/16) Calculation steps

z7 = ((-5i)^(1/8))*(8^(1/3)) = -0.4771303-2.3986959i = 2.4456891 × ei (-9π/16) Calculation steps

z8 = ((-5i)^(1/8))*(8^(1/3)) = 1.3587521-2.0335162i = 2.4456891 × ei (-5π/16) Calculation steps

z9 = ((-5i)^(1/8))*(8^(1/3)) = -0.786141+2.3158967i = 2.4456891 × ei 29π/48 Calculation steps

z10 = ((-5i)^(1/8))*(8^(1/3)) = -2.1934719+1.0817006i = 2.4456891 × ei 41π/48 Calculation steps

z11 = ((-5i)^(1/8))*(8^(1/3)) = -2.3158967-0.786141i = 2.4456891 × ei (-43π/48) Calculation steps

z12 = ((-5i)^(1/8))*(8^(1/3)) = -1.0817006-2.1934719i = 2.4456891 × ei (-31π/48) Calculation steps

z13 = ((-5i)^(1/8))*(8^(1/3)) = 0.786141-2.3158967i = 2.4456891 × ei (-19π/48) Calculation steps

z14 = ((-5i)^(1/8))*(8^(1/3)) = 2.1934719-1.0817006i = 2.4456891 × ei (-7π/48) Calculation steps

z15 = ((-5i)^(1/8))*(8^(1/3)) = 2.3158967+0.786141i = 2.4456891 × ei 0.3272492 = 2.4456891 × ei 0.1041667 π Calculation steps

z16 = ((-5i)^(1/8))*(8^(1/3)) = 1.0817006+2.1934719i = 2.4456891 × ei 17π/48 Calculation steps

z17 = ((-5i)^(1/8))*(8^(1/3)) = -1.6125549-1.8387664i = 2.4456891 × ei (-35π/48) Calculation steps

z18 = ((-5i)^(1/8))*(8^(1/3)) = 0.1599557-2.4404527i = 2.4456891 × ei (-23π/48) Calculation steps

z19 = ((-5i)^(1/8))*(8^(1/3)) = 1.8387664-1.6125549i = 2.4456891 × ei (-11π/48) Calculation steps

z20 = ((-5i)^(1/8))*(8^(1/3)) = 2.4404527+0.1599557i = 2.4456891 × ei 0.0654498 = 2.4456891 × ei 0.0208333 π Calculation steps principal root

z21 = ((-5i)^(1/8))*(8^(1/3)) = 1.6125549+1.8387664i = 2.4456891 × ei 13π/48 Calculation steps

z22 = ((-5i)^(1/8))*(8^(1/3)) = -0.1599557+2.4404527i = 2.4456891 × ei 25π/48 Calculation steps

z23 = ((-5i)^(1/8))*(8^(1/3)) = -1.8387664+1.6125549i = 2.4456891 × ei 37π/48 Calculation steps

z24 = ((-5i)^(1/8))*(8^(1/3)) = -2.4404527-0.1599557i = 2.4456891 × ei (-47π/48) Calculation steps