A power such as 2⁵ has a base, 2, and an index, 5, which says how many times the base is multiplied by itself. When the bases are the same, multiplying means adding the indices, dividing means subtracting them, and a power of a power means multiplying them. Anything to the power 0 is 1.
Laws of indices
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (for a ≠ 0)
- a¹ = a
The laws only apply when the bases are the same.
Reading a power
In 3⁵, the base is 3 and the index is 5. It means 3 × 3 × 3 × 3 × 3 = 243: five 3s multiplied together, not 3 × 5. With letters, x⁴ means x × x × x × x.
Multiplying and dividing powers
When the bases are the same, multiplying powers adds the indices: x³ × x⁴ = x⁷, because there are seven xs multiplied altogether. Dividing subtracts the indices: x⁹ ÷ x³ = x⁶. If there are numbers in front, deal with them separately: 3x² × 5x⁴ = 15x⁶.
A power of a power
When a power is raised to another power, multiply the indices: (x²)³ = x⁶, because it means x² × x² × x². If a bracket contains a number as well, raise the number to the power too: (2x³)² = 4x⁶.
The zero index
Any non-zero number to the power 0 equals 1. This follows from the division law: x⁵ ÷ x⁵ = x⁰, and anything divided by itself is 1. So 7⁰ = 1 and (3y)⁰ = 1.
Finding a missing index
Some questions give an equation and ask for a missing power, such as 3ˣ × 3⁴ = 3¹⁰. Use the laws to combine the left-hand side into a single power, 3ˣ⁺⁴, then compare the indices: x + 4 = 10, so x = 6. If the numbers are not written as powers of the same base, rewrite them first: 8 = 2³ and 27 = 3³.
Simplifying expressions
Apply the laws one step at a time, working from left to right, and treat the numbers and each letter separately. Write the final answer as a single power where the question asks for it.
Worked examples
Example 1. Work out the value of 5³.
- 5 × 5 × 5.
- 5 × 5 = 25, then 25 × 5.
Answer: 125.
Example 2. Simplify p⁶ × p⁵ ÷ p².
- Multiply: add the indices, 6 + 5 = 11, giving p¹¹.
- Divide: subtract, 11 − 2 = 9.
Answer: p⁹.
Example 3. Simplify 3x² × 5x⁴.
- Numbers: 3 × 5 = 15.
- Powers of x: add the indices, 2 + 4 = 6.
Answer: 15x⁶.
Example 4. Simplify (2x³)².
- Square the number: 2² = 4.
- Power of a power: x³ˣ² = x⁶.
Answer: 4x⁶.
Example 5. Work out 5⁰ + 3² × 3.
- 5⁰ = 1.
- 3² × 3 = 3³ = 27.
- 1 + 27.
Answer: 28.
Common mistakes
- Multiplying the base by the index. 5³ = 125, not 15.
- Multiplying indices when multiplying powers. x³ × x⁴ = x⁷, not x¹².
- Forgetting the number in a bracket. (2x³)² = 4x⁶, not 2x⁶.
- Thinking a⁰ = 0. Any non-zero number to the power 0 is 1.
- Combining different bases. 2³ × 3² cannot be written as a single power.
How index laws are examined
Index laws appear on both tiers of the 1MA1 GCSE and the 4MA1 International GCSE, usually as short simplifying questions, and at Higher tier they extend to negative and fractional indices. Questions ask you to:
- write a product or quotient as a single power
- simplify algebraic expressions using the laws
- use the zero index
- find a missing index in an equation such as 2ˣ × 2³ = 2¹⁰
The indices worksheet includes full answers, and the GCSE number revision engine generates more questions.
Next, see negative and fractional indices. The same laws are used in standard form calculations.
Frequently asked questions
What are the laws of indices?
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ and a⁰ = 1.
Why is anything to the power 0 equal to 1?
Because aⁿ ÷ aⁿ = a⁰, and any non-zero number divided by itself is 1.
Can you add indices for different bases?
No. The laws only work when the bases are the same.