An example

An example is playfully for grade 6 from Math, and I don't know how to explain it to my daughter when I don't want to use the calculator to calculate the cube root. Thus:

The student made a cuboid from a block of 16x18x48 mm of plasticine. What will be the edge of the cube? If I calculate the cuboid volume, would I have to cube root or not? Thank you for your answer.

Correct answer:

a₂ =  24 mm

Step-by-step explanation:

$$\begin{gathered} a=16\text{mm} \\ b=18\text{mm} \\ c=48\text{mm} \\ V=a\cdot b\cdot c=16\cdot 18\cdot 48=13824{\text{mm}}^3 \\ 16=2^4 \\ 18=2\cdot 3^2 \\ 48=2^4\cdot 3 \\ 13824=2^9\times 3^3=(2^3{)}^3\cdot 3^3=8^3\cdot 3^3=(8\cdot 3{)}^3=24^3 \\ V=a_2^3 \\ {x}_0=\frac{a+b+c}{3}=\frac{16+18+48}{3}=\frac{82}{3}\doteq 27.3333\text{mm} \\ {V}_0=27^3=19683{\text{mm}}^3 \\ {V}_1=28^3=21952{\text{mm}}^3 \\ {V}_3=26^3=17576{\text{mm}}^3 \\ {V}_4=25^3=15625{\text{mm}}^3 \\ {V}_5=24^3=13824{\text{mm}}^3 \\ {V}_5=V \\ {a}_2=24\text{mm} \\ \text{ Verifying Solution: } \\ 13824=2^9\cdot 3^3=(2^3\cdot 3{)}^3=(8\cdot 3{)}^3=24^3 \\ {a}_2=24\text{mm} \\ {a}_2=\sqrt[3]{V}=\sqrt[3]{13824}=24\text{mm} \end{gathered}$$

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