Index Laws
Grade 4What index notation means
When we write 5³, the 5 is the base and the 3 is the index (also called a power or exponent). It just means 5 × 5 × 5. Index laws are rules that let you simplify expressions involving the same base without expanding everything out.
The three laws
Multiplying — add the powers. When you multiply two powers of the same base, add the indices.
Dividing — subtract the powers. When you divide two powers of the same base, subtract the indices.
Power of a power — multiply the powers. When you raise a power to another power, multiply the indices.
There is also a fourth rule worth knowing: any base raised to the power zero equals 1. So 7⁰ = 1, and even 1000⁰ = 1.
Worked examples
Don't get these mixed up
Try this one
Without using a calculator, simplify fully \( \dfrac{4^{3} \times 8^{2}}{2^{10}} \).
Give your answer as a power of 2. [3]
This uses index laws with the same base.
Ask yourself: 'can I write every number in the expression as a power of 2?' Once all bases match, use the multiplication and division index laws: \( a^m \times a^n = a^{m+n} \) and \( a^m \div a^n = a^{m-n} \).
