YEAR 10 · EXPONENTIAL RELATIONSHIPS · 15 MIN

Integer and fractional index laws

Use index laws to simplify products, quotients and powers, including negative and fractional indices.

In this lesson

Select an index law from the operation
State sensible restrictions on the base

Before you start

Year 9 index laws

01 / LEARN

Understand the idea.

Identify the operation

First identify the operation between the powers. With a common base, a product adds indices and a quotient subtracts them: $a^{m} a^{n} = a^{m + n}$ and $\frac{a^{m}}{a^{n}} = a^{m - n}$ , with $a \neq 0$ for division.

A power of a power multiplies indices: ${(a^{m})}^{n} = a^{m n}$ . Compare ${(a^{3})}^{5} = a^{15}$ with $a^{3} a^{5} = a^{8}$ : the brackets change which operation you perform. Addition of powers has no matching shortcut.

Zero, negative and fractional powers

For $a \neq 0$ , $a^{0} = 1$ and $a^{- n} = \frac{1}{a^{n}}$ . A negative index takes a reciprocal; it does not attach a minus sign to the answer. For instance, $2^{- 3} = \frac{1}{8}$ .

For $a > 0$ and positive integer $n$ , $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ . The denominator names the root and the numerator names the power. Using positive bases here keeps these index laws within the real numbers.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Combine two laws

Simplify $\frac{x^{7} x^{- 2}}{x^{3}}$ , where $x \neq 0$ .

Worked solution

1
Combine the numerator powers.
$x^{7} x^{- 2} = x^{7 + (- 2)} = x^{5}$
Multiplication of equal bases adds their indices.
2
Divide the remaining powers.
$\frac{x^{5}}{x^{3}} = x^{5 - 3}$
The nonzero-base condition permits subtraction of indices in a quotient.
3
Simplify the exponent.
$\frac{x^{7} x^{- 2}}{x^{3}} = x^{2} (x \neq 0)$
The original restriction remains even though the final expression has no denominator.

Answer

$x^{2}$

EXAMPLE 02

Evaluate a fractional power

Evaluate $16^{3 / 4}$ .

Worked solution

1
Read the fractional index.
$16^{\frac{3}{4}} = {(\sqrt[4]{16})}^{3}$
The denominator gives a fourth root and the numerator gives a cube.
2
Evaluate the root.
$\sqrt[4]{16} = 2$
The principal fourth root is positive and $2^{4} = 16$ .
3
Cube the root.
$16^{\frac{3}{4}} = 2^{3} = 8$
Taking the root first keeps the arithmetic small.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify ${(b^{2})}^{3} b^{- 1}$ for $b \neq 0$ .

Show answer & reasoning

$b^{5}$

First ${(b^{2})}^{3} = b^{6}$ , then $b^{6} b^{- 1} = b^{6 - 1} = b^{5}$ . The condition $b \neq 0$ is needed for the negative power.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Simplify $x^{4} x^{6}$ .
2 marks
Simplify $\frac{a^{8}}{a^{3}}$ , $a \neq 0$ .
2 marks
Simplify ${(m^{3})}^{4}$ .
2 marks
Evaluate $5^{- 2}$ .
2 marks
Evaluate $81^{\frac{3}{4}}$ .
2 marks
Simplify $\frac{p^{- 2} p^{5}}{p^{4}}$ for $p \neq 0$ , using positive indices.
3 marks

Read the worked solutions

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

Simplify $x^{4} x^{6}$ .
2 marks
Add indices for the common-base product: $4 + 6 = 10$ .
$x^{10}$
Simplify $\frac{a^{8}}{a^{3}}$ , $a \neq 0$ .
2 marks
$8 - 3 = 5$ .
$a^{5}$
Simplify ${(m^{3})}^{4}$ .
2 marks
Power of a power: $3 \times 4 = 12$ .
$m^{12}$
Evaluate $5^{- 2}$ .
2 marks
$5^{- 2} = \frac{1}{5^{2}} = \frac{1}{25}$ .
$\frac{1}{25}$
Evaluate $81^{\frac{3}{4}}$ .
2 marks
$81^{\frac{1}{4}} = 3$ , so $81^{\frac{3}{4}} = 3^{3} = 27$ .
27
Simplify $\frac{p^{- 2} p^{5}}{p^{4}}$ for $p \neq 0$ , using positive indices.
3 marks
$- 2 + 5 - 4 = - 1$ , so $p^{- 1} = \frac{1}{p}$ .
$\frac{1}{p}$

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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.

Identify the operation

Zero, negative and fractional powers

See the method in action.

Combine two laws

Combine the numerator powers.

Divide the remaining powers.

Simplify the exponent.

Evaluate a fractional power

Read the fractional index.

Evaluate the root.

Cube the root.

Try it yourself.

Check your understanding

Keep practising