Rules of Indices (Exponents)

Index Law 1: Product of Powers

When multiplying powers with the same base, add the indices: \( a^m \times a^n = a^{m+n} \)

Example: \( 2^3 \times 2^4 = 2^{3+4} = 2^7 \)

Index Law 2: Quotient of Powers

When dividing powers with the same base, subtract the indices: \( a^m \div a^n = a^{m – n} \)

Example: \( 5^6 \div 5^2 = 5^{6-2} = 5^4 \)

Index Law 3: Power of a Power

When raising a power to another power, multiply the indices: \( (a^m)^n = a^{mn} \)

Example: \( (3^2)^4 = 3^{2 \times 4} = 3^8 \)

Index Law 4: Power of a Product

Apply the index to each factor: \( (ab)^n = a^n \times b^n \)

Example: \( (2 \times 3)^4 = 2^4 \times 3^4 = 16 \times 81 = 1296 \)

Index Law 5: Power of a Quotient

Apply the index to both numerator and denominator: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)

Example: \( \left( \frac{4}{5} \right)^2 = \frac{4^2}{5^2} = \frac{16}{25} \)

Index Law 6: Zero Index

Any non-zero number to the power 0 is 1: \( a^0 = 1 \), for \( a \ne 0 \)

Index Law 7: Negative Indices

A negative index represents a reciprocal: \( a^{-n} = \frac{1}{a^n} \), for \( a \ne 0 \)

Important Notes

• These laws only apply when the base is the same.
• Always simplify expressions before applying the laws.