Year 8 - Exponents - Free practice

Year 8 Exponents (Index Laws) Worksheets

Multiply, divide, power-of-a-power - the four index laws.

Quick answer

What are the exponent laws for multiplying and dividing powers?

When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$ . When dividing, subtract them: $a^m \div a^n = a^{m-n}$ . Raising a power to a power means multiplying exponents: $(a^m)^n = a^{mn}$ . Any base to the power of zero equals 1: $a^0 = 1$ . For example, $2^3 \times 2^4 = 2^7 = 128$ .

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Worked examples

Year 8 exponents: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Multiplying and dividing with the same base
Simplify each expression. Give your answer in exponent form.

(a) $3^4 \times 3^5$

(b) $\dfrac{2^9}{2^3}$
Show worked solution
Key idea: When multiplying powers with the same base, add the exponents. When dividing, subtract the exponents.
1. (a) $3^4 \times 3^5 = 3^{4+5} = 3^9$
2. (b) $\dfrac{2^9}{2^3} = 2^{9-3} = 2^6$
Answer: (a) $3^9$ (b) $2^6$
Example 2 · Mixed rules including power-of-a-power and zero index
Simplify each expression.

(a) $(5^3)^4$

(b) $\dfrac{(2^3)^4 \times 2^0}{2^5}$
Show worked solution
Key idea: For a power of a power, multiply the exponents. Any base raised to the power of zero equals 1.
1. (a) $(5^3)^4 = 5^{3 \times 4} = 5^{12}$
2. (b) Simplify the numerator first: $(2^3)^4 = 2^{12}$ and $2^0 = 1$ , so the numerator is $2^{12} \times 1 = 2^{12}$ .
3. Now divide: $\dfrac{2^{12}}{2^5} = 2^{12-5} = 2^7$
Answer: (a) $5^{12}$ (b) $2^7$
Example 3 · Exponent laws in a real-life context
A factory produces $6^8$ small bolts on Monday and uses $6^3$ bolts to fill each crate. How many full crates can the factory fill? Write your answer as a power of 6.
Show worked solution
Key idea: Dividing quantities with the same base means subtracting the exponents.
1. Number of crates $= \dfrac{6^8}{6^3}$
2. Apply the division law: $6^8 \div 6^3 = 6^{8-3} = 6^5$
3. Check: $6^5 \times 6^3 = 6^{5+3} = 6^8$ ✓
Answer: $6^5$ crates

Common mistakes

Where students usually lose marks

Multiplying the bases instead of adding the exponents

$3^4 \times 3^5 = 9^9$

When multiplying powers with the same base, keep the base and add the exponents: $3^4 \times 3^5 = 3^{4+5} = 3^9$ . The base never changes.

Multiplying exponents instead of adding them

$2^3 \times 2^4 = 2^{12}$

Multiplying exponents is the rule for a power of a power, such as $(2^3)^4 = 2^{12}$ . For $2^3 \times 2^4$ , you add the exponents: $2^{3+4} = 2^7$ .

Thinking a⁰ = 0

$7^0 = 0$

Any non-zero number raised to the power of zero equals 1, not 0. For example, $7^0 = 1$ . You can see why from the division law: $7^3 \div 7^3 = 7^{3-3} = 7^0$ , and any number divided by itself is 1.

Subtracting exponents in the wrong order when dividing

$\dfrac{5^7}{5^3} = 5^{3-7} = 5^{-4}$

Always subtract the exponent of the denominator from the exponent of the numerator: $\dfrac{5^7}{5^3} = 5^{7-3} = 5^4$ .

Try one yourself

Your turn

Simplify the following expression and write your answer as a single power of $3$ . $\frac{(3^4)^3}{3^5 \times 3^2}$

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About this worksheet

Why we built it

Index laws ("the exponent rules") trip up students because there are several of them and they look similar. These worksheets isolate each rule and then mix them up so your student learns to recognise which law applies before they start writing.

What's covered

Sub-skills your student will practise

✓Multiplying terms with the same base (a^m x a^n = a^(m+n))
✓Dividing terms with the same base (a^m / a^n = a^(m-n))
✓Power-of-a-power rule
✓Zero index (a^0 = 1)
✓Mixed-rule simplification problems

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Frequently asked

Questions parents ask about exponents

What are the four index laws?

(1) a^m x a^n = a^(m+n) when multiplying. (2) a^m / a^n = a^(m-n) when dividing. (3) (a^m)^n = a^(m x n) for power of a power. (4) a^0 = 1 for any non-zero base.

Why does anything to the power of zero equal one?

Because of the division law: a^n / a^n = a^(n-n) = a^0. But anything divided by itself is also 1. So a^0 = 1. This is the cleanest justification for an otherwise mysterious rule.

What gets students confused about index laws?

Adding exponents when they should be multiplying. (a^3)^2 is a^6, not a^5 - the rule is multiplication of exponents for a power of a power, not addition. Worksheets that mix all four laws are the only way to make this stick.

Are negative exponents Year 8?

Year 8 stays with integer and zero exponents. Negative exponents (a^-1 = 1/a) are formally Year 9, though many teachers preview them once the four basic laws are solid.

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