1.8 Perfect Cubes and Cube Roots

A perfect cube works the same way, in threes: $0, 1, 8, 27, 64, 125, \ldots$ In $3375 = 3^3 \times 5^3$ every index is a multiple of $3$ , so the factors split into three identical groups, and dividing each index by $3$ gives the root.

Perfect cube test, and its root

\text{every index a multiple of } 3 \iff \text{perfect cube}, \qquad \sqrt[3]{a^p \times b^q} = a^{p/3} \times b^{q/3}

Finding a cube root using prime factorisation

Given that $13824 = 2^9 \times 3^3$ , find $\sqrt[3]{13824}$ .

$13824 = 2^9 \times 3^3 \quad \scriptsize\textit{prime factorisation}$
All indices are multiples of $3$ ( $9$ and $3$ ), so $13824$ is a perfect cube.
$\sqrt[3]{13824} = 2^3 \times 3 \quad \scriptsize\textit{divide each index by 3}$
$= 8 \times 3$
$= 24$

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A practice question on this appears here. Practise primes hcf lcm.

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