Integer and fractional index laws | Year 10

01 / LEARN

The idea, step by step

Identify the operation

For a common base, multiplication adds exponents and division subtracts them. A power raised to another power multiplies the exponents. The operation between the powers decides the law.

For example, $a^{3} a^{5} = a^{8}$ , but ${(a^{3})}^{5} = a^{15}$ . If the expression is a sum, neither of these shortcuts applies.

Zero, negative and fractional powers

For $a \neq 0$ , $a^{0} = 1$ and $a^{- n} = 1 / a^{n}$ . A negative exponent changes the position of a factor; it does not make the value negative.

For positive real $a$ , $a^{m / n}$ means the $n$ th root raised to the power $m$ . This positive-base setting lets us apply the familiar index laws without introducing invalid real roots.

A common trap

$a^{3} + a^{4} = a^{7}$ .

Instead: Only a product of equal bases adds indices. The sum stays $a^{3} + a^{4}$ , or factors to $a^{3} (1 + a)$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Combine two laws

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

The product in the numerator is $x^{7 - 2} = x^{5}$ .
Divide equal bases by subtracting indices: $x^{5 - 3} = x^{2}$ .

x^{2}

Example 2

Evaluate a fractional power

Evaluate $16^{3 / 4}$ .

The fourth root of $16$ is $2$ .
Cube that root: $2^{3} = 8$ .

Check your understanding

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

Show answer

$b^{5}$

Multiply inside the power, then add: $2 \times 3 - 1 = 5$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Simplify $x^{4} x^{6}$ .
2 marks
Simplify $a^{8} / a^{3}$ , $a \neq 0$ .
2 marks
Simplify ${(m^{3})}^{4}$ .
2 marks
Evaluate $5^{- 2}$ .
2 marks
Evaluate $81^{3 / 4}$ .
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 $a^{8} / a^{3}$ , $a \neq 0$ .
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
$5^{- 2} = 1 / 5^{2} = 1 / 25$ .
$1 / 25$
Evaluate $81^{3 / 4}$ .
2 marks
$81^{1 / 4} = 3$ , so $81^{3 / 4} = 3^{3} = 27$ .
27
Simplify $\frac{p^{- 2} p^{5}}{p^{4}}$ for $p \neq 0$ , using positive indices.
3 marks
$- 2 + 5 - 4 = - 1$ , so $p^{- 1} = 1 / p$ .
$1 / p$

Checking saved progress…

Continue learning

Connecting fractional powers with roots

Next lesson

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 ↗

Back to Exponential relationships