Integer and fractional index laws | Year 11

01 / LEARN

The idea, step by step

Read the exponent first

A positive integer exponent counts equal factors. A negative exponent reverses multiplication: $a^{- n} = 1 / a^{n}$ , where $a \neq 0$ . It does not make the result negative.

For a positive base, a fractional exponent uses its denominator as a root and its numerator as a power. Take an exact root first when it makes the arithmetic easier.

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

Combine structure before calculating

For equal non-zero bases, multiplication adds exponents and division subtracts exponents. Raising a power to another power multiplies exponents; keep brackets around a whole product.

Restrictions matter: $\sqrt{x^{2}} = ∣ x ∣$ , not always $x$ . Our fractional-power variable examples state $x > 0$ so that root and power rules can be used without sign ambiguity.

a^{m} a^{n} = a^{m + n}, {(a^{m})}^{n} = a^{m n}

A common trap

Writing $a^{- 2} = - a^{2}$ .

Instead: A negative exponent means reciprocal: $a^{- 2} = 1 / a^{2}$ for $a \neq 0$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A fractional power

Evaluate $81^{3 / 4}$ .

$\sqrt[4]{81} = 3$ .
Cube the root: $3^{3} = 27$ .

Example 2

A powered product

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

Apply the exponent to both factors: $16^{3 / 4} x^{8 (3 / 4)}$ .
$16^{3 / 4} = 8$ and $8 (3 / 4) = 6$ .

8 x^{6}

Check your understanding

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

Show answer

$1 / 9$

Cube root 27 to get 3, square to get 9, then take the reciprocal.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Evaluate $2^{- 3}$ .
2 marks
Evaluate $32^{2 / 5}$ .
2 marks
Evaluate $16^{- 3 / 4}$ .
2 marks
Simplify $a^{4} a^{3} / a^{2}$ for $a \neq 0$ .
2 marks
Simplify ${(16 x^{4})}^{3 / 4}$ for $x > 0$ .
3 marks
Check the claim ${(9 + 16)}^{1 / 2} = 9^{1 / 2} + 16^{1 / 2}$ 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
$2^{- 3} = 1 / 2^{3} = 1 / 8$ .
$1 / 8$
Evaluate $32^{2 / 5}$ .
2 marks
$\sqrt[5]{32} = 2$ , then $2^{2} = 4$ .
4
Evaluate $16^{- 3 / 4}$ .
2 marks
$\sqrt[4]{16} = 2$ ; cube to get 8 and invert.
$1 / 8$
Simplify $a^{4} a^{3} / a^{2}$ for $a \neq 0$ .
2 marks
The exponent is $4 + 3 - 2 = 5$ .
$a^{5}$
Simplify ${(16 x^{4})}^{3 / 4}$ for $x > 0$ .
3 marks
$16^{3 / 4} = 8$ ; $4 (3 / 4) = 3$ .
$8 x^{3}$
Check the claim ${(9 + 16)}^{1 / 2} = 9^{1 / 2} + 16^{1 / 2}$ 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.

Checking saved progress…

Continue learning

Rational/irrational numbers and surd operations

Next lesson

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 ↗

Back to Exponentials and logarithms