Index law rules

Apply the product rule, quotient rule and power of a power rule to simplify expressions with indices.

Worked examples

Using the product and quotient rules

Straightforward

Problem

Simplify

7^5 \times 7^3 \div 7^4

. Give your answer as a power of

7

Using the power of a power rule with the quotient rule

Moderate

Problem

Simplify

(4^3)^2 \div 4^5

. Give your answer as a power of

4

Combining all three index laws

Challenging

Problem

Simplify

\dfrac{(6^2)^3 \times 6^4}{6^3 \times 6^5}

. Give your answer as a power of

6

Practise

Q1·Straightforward

Simplify

3^4 \times 3^2

using the product rule. Give your answer as

3^n

. What is the value of

n

Q2·Straightforward

Simplify

2^7 \div 2^3

using the quotient rule. Give your answer as

2^n

. What is the value of

n

Q3·Straightforward

Simplify

5^3 \times 5^5

using the product rule. Give your answer as

5^n

. What is the value of

n

Q4·Straightforward

Simplify

6^9 \div 6^4

using the quotient rule. Give your answer as

6^n

. What is the value of

n

Q5·Moderate

Simplify

(2^3)^4

using the power of a power rule. Give your answer as

2^n

. What is the value of

n

Q6·Moderate

Simplify

(5^2)^3

using the power of a power rule. Give your answer as

5^n

. What is the value of

n

Q7·Moderate

Simplify

3^2 \times 3^4 \div 3^3

using the index laws. Give your answer as

3^n

. What is the value of

n

Q8·Moderate

Simplify

(7^3)^2 \div 7^4

using the index laws. Give your answer as

7^n

. What is the value of

n

Q9·Challenging

Simplify

2^3 \times (2^2)^4 \div 2^5

using the index laws. Give your answer as

2^n

. What is the value of

n

Q10·Challenging

Simplify

(3^4)^2 \times 3^3 \div 3^7

using the index laws. Give your answer as

3^n

. What is the value of

n

Q11·Challenging

Simplify

\dfrac{5^6 \times (5^2)^3}{5^4}

using the index laws. Give your answer as

5^n

. What is the value of

n

Q12·Challenging

Simplify

(4^3 \times 4^2)^2 \div 4^6

using the index laws. Give your answer as

4^n

. What is the value of

n