Connecting fractional powers with roots | Year 10

01 / LEARN

The idea, step by step

The denominator names the root

In $a^{m / n}$ , the denominator $n$ describes a root and the numerator $m$ describes a power. Thus $a^{1 / 2} = \sqrt{a}$ and $a^{1 / 3} = \sqrt[3]{a}$ .

For a positive base you may take the root first or the power first. Choose the root first when it keeps the arithmetic small.

Real values and order matter

An even root is real only when its radicand is nonnegative. We use the principal, nonnegative square root. For example, $\sqrt{25} = 5$ , even though both $5$ and $- 5$ square to $25$ .

Odd roots can accept negative inputs: $\sqrt[3]{- 8} = - 2$ . Be careful about applying unrestricted index-law shortcuts to negative bases; use the stated root interpretation.

A common trap

$36^{1 / 2} = 18$ because the index says divide by $2$ .

Instead: The index $1 / 2$ means square root, not division. $36^{1 / 2} = 6$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Root then power

Evaluate $64^{2 / 3}$ .

The cube root of $64$ is $4$ .
Square the result: $4^{2} = 16$ .

Example 2

A reciprocal root

Evaluate $25^{- 1 / 2}$ .

The square root is $25^{1 / 2} = 5$ .
The negative index takes the reciprocal: $1 / 5$ .

1 / 5

Check your understanding

Write $\sqrt[5]{x^{3}}$ as a fractional power for $x > 0$ .

Show answer

$x^{3 / 5}$

The fifth root contributes denominator $5$ and the cube contributes numerator $3$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write $\sqrt{x}$ as a power, where $x \geq 0$ .
2 marks
Write $a^{4 / 3}$ in root notation for $a > 0$ .
2 marks
Evaluate $125^{2 / 3}$ .
2 marks
Evaluate $16^{- 3 / 4}$ .
2 marks
Evaluate $\sqrt[3]{- 216}$ .
2 marks
Explain why ${(- 9)}^{1 / 2}$ is not a real number.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $\sqrt{x}$ as a power, where $x \geq 0$ .
2 marks
A square root has index $1 / 2$ .
$x^{1 / 2}$
Write $a^{4 / 3}$ in root notation for $a > 0$ .
2 marks
Take the cube root, then raise it to the fourth power.
${(\sqrt[3]{a})}^{4}$
Evaluate $125^{2 / 3}$ .
2 marks
$\sqrt[3]{125} = 5$ , then $5^{2} = 25$ .
25
Evaluate $16^{- 3 / 4}$ .
2 marks
$16^{1 / 4} = 2$ , $2^{3} = 8$ , and the negative power gives $1 / 8$ .
$1 / 8$
Evaluate $\sqrt[3]{- 216}$ .
2 marks
${(- 6)}^{3} = - 216$ .
−6
Explain why ${(- 9)}^{1 / 2}$ is not a real number.
3 marks
If $u$ is real then $u^{2} \geq 0$ , so no real $u$ has $u^{2} = - 9$ .
No real square is negative.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

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