YEAR 10 · EXPONENTIAL RELATIONSHIPS · 15 MIN

Connecting fractional powers with roots

Read fractional indices as roots and move between the two notations.

In this lesson

Convert roots into fractional powers
Evaluate positive-base powers efficiently

Before you start

Squares and cubes
Integer index laws

01 / LEARN

Understand the idea.

The denominator names the root

In $a^{\frac{m}{n}}$ , read the denominator as a root and the numerator as a power: $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ for $a > 0$ . Thus $a^{\frac{1}{2}} = \sqrt{a}$ and $a^{\frac{1}{3}} = \sqrt[3]{a}$ .

Choose the order that makes evaluation easier. For $64^{\frac{2}{3}}$ , finding the cube root first avoids squaring $64$ . A negative fractional index adds one more action: take the reciprocal of the corresponding positive power.

Real values and order matter

An even root requires a nonnegative radicand for a real answer. The square-root sign selects the principal root: $\sqrt{25} = 5$ . Solving the different question $x^{2} = 25$ gives both $x = 5$ and $x = - 5$ .

Odd roots can be negative: $\sqrt[3]{- 8} = - 2$ because ${(- 2)}^{3} = - 8$ . For negative bases, keep the stated root interpretation rather than applying fractional-index rules without checking whether every expression is real.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Root then power

Evaluate $64^{2 / 3}$ .

Worked solution

1
Translate the index into a root and power.
$64^{\frac{2}{3}} = {(\sqrt[3]{64})}^{2}$
The denominator $3$ specifies a cube root.
2
Find the cube root.
$\sqrt[3]{64} = 4$
Cubing $4$ gives $64$ .
3
Apply the numerator power.
$64^{\frac{2}{3}} = 4^{2} = 16$
The numerator $2$ squares the root.

Answer

EXAMPLE 02

A reciprocal root

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

Worked solution

1
Remove the negative index.
$25^{- \frac{1}{2}} = \frac{1}{25^{\frac{1}{2}}}$
A negative index means the reciprocal of the positive power.
2
Evaluate the denominator.
$25^{\frac{1}{2}} = \sqrt{25} = 5$
A power of one half is the principal square root.
3
Write the exact value.
$25^{- \frac{1}{2}} = \frac{1}{5}$
The reciprocal remains positive because the base is positive.

Answer

$1 / 5$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$x^{3 / 5}$

The cube supplies numerator $3$ and the fifth root supplies denominator $5$ : $\sqrt[5]{x^{3}} = x^{\frac{3}{5}}$ for the stated $x > 0$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Write $\sqrt{x}$ as a power, where $x \geq 0$ .
2 marks
Write $a^{\frac{4}{3}}$ in root notation for $a > 0$ .
2 marks
Evaluate $125^{\frac{2}{3}}$ .
2 marks
Evaluate $16^{- \frac{3}{4}}$ .
2 marks
Evaluate $\sqrt[3]{- 216}$ .
2 marks
Explain why ${(- 9)}^{\frac{1}{2}}$ 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 $\frac{1}{2}$ .
$x^{\frac{1}{2}}$
Write $a^{\frac{4}{3}}$ in root notation for $a > 0$ .
2 marks
Take the cube root, then raise it to the fourth power.
${(\sqrt[3]{a})}^{4}$
Evaluate $125^{\frac{2}{3}}$ .
2 marks
$\sqrt[3]{125} = 5$ , then $5^{2} = 25$ .
25
Evaluate $16^{- \frac{3}{4}}$ .
2 marks
$16^{\frac{1}{4}} = 2$ , $2^{3} = 8$ , and the negative power gives $\frac{1}{8}$ .
$\frac{1}{8}$
Evaluate $\sqrt[3]{- 216}$ .
2 marks
${(- 6)}^{3} = - 216$ .
−6
Explain why ${(- 9)}^{\frac{1}{2}}$ 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 ↗

In this lesson

Understand the idea.

The denominator names the root

Real values and order matter

See the method in action.

Root then power

Translate the index into a root and power.

Find the cube root.

Apply the numerator power.

A reciprocal root

Remove the negative index.

Evaluate the denominator.

Write the exact value.

Try it yourself.

Check your understanding

Keep practising