1.8Perfect 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

\displaystyle \text{Perfect cube exactly when every index is a multiple of } 3, \qquad \sqrt[3]{a^p \times b^q} = a^{p/3} \times b^{q/3}

Nine splits into three groups of three, and the cube root takes one factor from each group.

Finding a cube root using prime factorisation

Given that $\displaystyle 13\,824 = 2^9 \times 3^3$ , find $\displaystyle \sqrt[3]{13\,824}$ .

Show solution▾

13\,824 = 2^9 \times 3^3 \quad \scriptsize\textit{prime factorisation}

All indices are multiples of

3

(

9

and

3

), so

13\,824

is a perfect cube.

\displaystyle \sqrt[3]{13\,824}

\displaystyle = 2^3 \times 3 \quad \scriptsize\textit{divide each index by 3}

\displaystyle = 8 \times 3

\displaystyle = 24

Now you try

Given that $15\,625 = 5^{6}$ , find $\sqrt[3]{15\,625}$ . Give your answer in index notation.

Show solution▾

\displaystyle \sqrt[3]{15\,625}

\displaystyle = \sqrt[3]{5^{6}}

\displaystyle = 5^{2} \quad \scriptsize\textit{divide each index by 3}

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