Integer and fractional index laws | Year 11

01 / LEARN

Understand the idea.

Read the exponent first

An exponent tells you how a base is used. A positive whole-number exponent repeats multiplication; a negative exponent takes the reciprocal of the corresponding positive power. It does not put a minus sign in front of the value.

For a positive base, the denominator of a fractional exponent specifies a root and the numerator specifies a power. When the root is exact, taking it first often avoids large intermediate numbers.

a^{m / n} = {(\sqrt[n]{a})}^{m} (a > 0)

Combine structure before calculating

Check the structure before applying an index law: multiplying equal bases adds exponents, dividing them subtracts exponents, and taking a power of a power multiplies exponents. Brackets show which complete expression the exponent acts on.

Sign restrictions are part of the method. The principal square root is nonnegative, so $\sqrt{x^{2}} = ∣ x ∣$ . The condition $x > 0$ in these fractional-power examples allows roots and powers to be simplified without sign ambiguity.

\begin{aligned} a^{m} a^{n} & = a^{m + n} \\ {(a^{m})}^{n} & = a^{m n} \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A fractional power

Evaluate $81^{3 / 4}$ .

Worked solution

1
Interpret the fractional exponent.
$81^{\frac{3}{4}} = {(\sqrt[4]{81})}^{3}$
The denominator gives the root and the numerator gives the subsequent power.
2
Take the exact fourth root.
$\sqrt[4]{81} = 3$
The positive fourth root is 3 because $3^{4} = 81$ .
3
Cube the result.
$81^{\frac{3}{4}} = 3^{3} = 27$
The numerator of the fractional exponent requires the third power.

Answer

EXAMPLE 02

A powered product

Simplify ${(16 x^{8})}^{3 / 4}$ for $x > 0$ .

Worked solution

1
Apply the exponent to both positive factors.
${(16 x^{8})}^{\frac{3}{4}} = 16^{\frac{3}{4}} {(x^{8})}^{\frac{3}{4}}$
The condition $x > 0$ makes the fractional-power product law valid here.
2
Evaluate the number and combine the variable powers.
$16^{\frac{3}{4}} = 2^{3} = 8, {(x^{8})}^{\frac{3}{4}} = x^{8 (\frac{3}{4})} = x^{6}$
A power of a power multiplies its exponents.
3
Combine the simplified factors.
${(16 x^{8})}^{\frac{3}{4}} = 8 x^{6}$
The numerical coefficient and variable factor are now both in simplest power form.

Answer

$8 x^{6}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Evaluate $27^{- 2 / 3}$ .

Show answer & reasoning

$1 / 9$

$27^{- \frac{2}{3}} = \frac{1}{{(\sqrt[3]{27})}^{2}} = \frac{1}{3^{2}} = \frac{1}{9}$ . The negative exponent requires the reciprocal after evaluating the positive power.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $2^{- 3}$ .
2 marks
Evaluate $32^{\frac{2}{5}}$ .
2 marks
Evaluate $16^{- \frac{3}{4}}$ .
2 marks
Simplify $\frac{a^{4} a^{3}}{a^{2}}$ for $a \neq 0$ .
2 marks
Simplify ${(16 x^{4})}^{\frac{3}{4}}$ for $x > 0$ .
3 marks
Check the claim ${(9 + 16)}^{\frac{1}{2}} = 9^{\frac{1}{2}} + 16^{\frac{1}{2}}$ and explain the error.
3 marks

Read the worked solutions

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

Evaluate $2^{- 3}$ .
2 marks
$2^{- 3} = \frac{1}{2^{3}} = \frac{1}{8}$ .
$\frac{1}{8}$
Evaluate $32^{\frac{2}{5}}$ .
2 marks
$\sqrt[5]{32} = 2$ , then $2^{2} = 4$ .
4
Evaluate $16^{- \frac{3}{4}}$ .
2 marks
$\sqrt[4]{16} = 2$ ; cube to get 8 and invert.
$\frac{1}{8}$
Simplify $\frac{a^{4} a^{3}}{a^{2}}$ for $a \neq 0$ .
2 marks
The exponent is $4 + 3 - 2 = 5$ .
$a^{5}$
Simplify ${(16 x^{4})}^{\frac{3}{4}}$ for $x > 0$ .
3 marks
$16^{\frac{3}{4}} = 8$ ; $4 (\frac{3}{4}) = 3$ .
$8 x^{3}$
Check the claim ${(9 + 16)}^{\frac{1}{2}} = 9^{\frac{1}{2}} + 16^{\frac{1}{2}}$ and explain the error.
3 marks
$\sqrt{25} = 5$ while $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$ .
A product power law does not apply to addition.
False: the sides are 5 and 7.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: exponential and logarithmic functions, growth models and numerical/algebraic prerequisites.

Official curriculum source ↗

In this lesson

Understand the idea.

Read the exponent first

Combine structure before calculating

See the method in action.

A fractional power

Interpret the fractional exponent.

Take the exact fourth root.

Cube the result.

A powered product

Apply the exponent to both positive factors.

Evaluate the number and combine the variable powers.

Combine the simplified factors.

Try it yourself.

Check your understanding

Keep practising