Write 3 × 3 × 3 × 3 as a single power of 3.

Count the repeated factors: 3 × 3 × 3 × 3 Use index notation: 3 × 3 × 3 × 3 = 3^4 State the answer: 3^4 Answer: 3^4

Simplify 5^3 × 5^4, giving your answer as a single power of 5.

Recall the product law: a^m × a^n = a^m+n Add the indices: 5^3 × 5^4 = 5^3+4 Write the single power: 5^3+4 = 5^7 Answer: 5^7

Simplify 7^9 ÷ 7^4, giving your answer as a single power of 7.

Recall the quotient law: a^m ÷ a^n = a^m-n Subtract the indices: 7^9 ÷ 7^4 = 7^9-4 Write the single power: 7^9-4 = 7^5 Answer: 7^5

Integer indices Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Integer indices questions. See exactly how to solve problems on index notation, evaluating powers, product law, quotient law.

index notationevaluating powersproduct lawquotient lawpower of a powerzero index

GCSE Foundation70 questionsStep-by-step solutions

Question 1

1 markeasy

Write

3 \times 3 \times 3 \times 3

as a single power of

3

Worked solution

Count the repeated factors
$3 \times 3 \times 3 \times 3$
The number $3$ is multiplied by itself, so count how many $3$ s there are. There are four of them.
Use index notation
$3 \times 3 \times 3 \times 3 = 3^4$
A power is a short way of writing repeated multiplication. The little number (the index) tells you how many $3$ s are multiplied together.
State the answer
$3^4$
So four $3$ s multiplied together is written as $3$ to the power $4$ .

Answer

3^4

Question 2

1 markeasy

Work out the value of

2^5

Worked solution

Write the power as repeated multiplication
$2^5 = 2 \times 2 \times 2 \times 2 \times 2$
The index $5$ tells you to multiply five $2$ s together. Writing them all out makes the calculation clear.
Multiply in stages
$2 \times 2 = 4,\ 4 \times 2 = 8,\ 8 \times 2 = 16$
Multiply one $2$ at a time and keep track of the running total. After four $2$ s you have $16$ .
Multiply by the last $2$
$16 \times 2 = 32$
One final doubling gives $32$ . So $2$ to the power $5$ is $32$ .

Answer

32

Question 3

2 markseasy

Simplify

5^3 \times 5^4

, giving your answer as a single power of

5

Worked solution

Recall the product law
$a^m \times a^n = a^{m+n}$
When you multiply powers of the same base, you add the indices. Both powers here have base $5$ , so the law applies.
Add the indices
$5^3 \times 5^4 = 5^{3+4}$
Three $5$ s multiplied by four more $5$ s gives seven $5$ s in total. That is why the indices are added.
Write the single power
$5^{3+4} = 5^7$
Adding $3$ and $4$ gives $7$ , so the answer is $5$ to the power $7$ .

Answer

5^7

Question 4

2 markseasy

Simplify

7^9 \div 7^4

, giving your answer as a single power of

7

Worked solution

Recall the quotient law
$a^m \div a^n = a^{m-n}$
When you divide powers of the same base, you subtract the indices. Both powers here have base $7$ .
Subtract the indices
$7^9 \div 7^4 = 7^{9-4}$
Dividing cancels four of the nine $7$ s on top, leaving five $7$ s. That is why the indices are subtracted.
Write the single power
$7^{9-4} = 7^5$
Subtracting $4$ from $9$ gives $5$ , so the answer is $7$ to the power $5$ .

Answer

7^5

Question 5

2 markseasy

Simplify

(2^3)^4

, giving your answer as a single power of

2

Worked solution

Recall the power-of-a-power law
$(a^m)^n = a^{mn}$
When a power is raised to another power, you multiply the indices. Here the inside power is $3$ and the outside power is $4$ .
Multiply the indices
$(2^3)^4 = 2^{3 \times 4}$
Raising $2$ cubed to the power $4$ means writing $2^3$ four times and multiplying, which gives $3 + 3 + 3 + 3 = 12$ twos in total.
Write the single power
$2^{3 \times 4} = 2^{12}$
Multiplying $3$ by $4$ gives $12$ , so the answer is $2$ to the power $12$ .

Answer

2^{12}

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