1.9Finding the Smallest k for Perfect Squares and Cubes

Given a number, find the smallest positive $k$ making the product, or the quotient, a perfect square or cube. Factorise, then fix whichever indices do not yet qualify.

Write the number in index notation.
For a square, every index must be even; for a cube, a multiple of $3$ .
Multiplying by $k$ : $k$ supplies whatever each short index still needs.
Dividing by $k$ : $k$ removes the surplus instead.

Finding smallest k for a perfect cube

Given that $\displaystyle 600 = 2^3 \times 3 \times 5^2$ , find the smallest positive integer $\displaystyle k$ such that $\displaystyle 600k$ is a perfect cube.

Show solution▾

600 = 2^3 \times 3^1 \times 5^2

For a perfect cube, each index must be a multiple of

3

Index of

2

3

: already a multiple of

3

. ✓

Index of

3

1

: it must reach

3

, so multiply by

3^2

Index of

5

2

: it must reach

3

, so multiply by

5^1

\displaystyle k

\displaystyle = 3^2 \times 5

\displaystyle = 45

Check:

600 \times 45 = 27\,000

= 2^3 \times 3^3 \times 5^3

= 30^3

✓

Dividing to make a perfect square

Given that $\displaystyle 8232 = 2^3 \times 3 \times 7^3$ , find the smallest positive integer $\displaystyle m$ such that $\displaystyle \frac{8232}{m}$ is a perfect square. Hence find $\displaystyle \sqrt{\frac{8232}{m}}$ .

Show solution▾

8232 = 2^3 \times 3^1 \times 7^3

For a perfect square, each index must be even.

Index of

2

3

(odd): divide by

2^1

to get

2^2

Index of

3

1

(odd): divide by

3^1

to get

3^0

Index of

7

3

(odd): divide by

7^1

to get

7^2

\displaystyle m

\displaystyle = 2 \times 3 \times 7

\displaystyle = 42

\displaystyle \frac{8232}{42}

\displaystyle = 196

\displaystyle = 2^2 \times 7^2

\displaystyle \sqrt{196}

\displaystyle = 2 \times 7

\displaystyle = 14

Now you try

Given that $128 = 2^{7}$ , find the smallest positive integer $h$ such that $\frac{128}{h}$ is a perfect cube.

Show solution▾

128 = 2^{7}

For a perfect cube, all indices must be divisible by

3

2^{7}

: remove 1 factor of

2

h = 2

Check:

\frac{128}{2} = 64

\displaystyle 64

\displaystyle = 2^{6} \quad \scriptsize\textit{all indices divisible by 3}

\displaystyle \sqrt[3]{64}

\displaystyle = 4 \checkmark

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