Laws of Indices (Core)

GCSE Algebra indices powers exponents
Practice this formula All formulas
\( a^m a^n = a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\frac{1}{a^n},\quad a^0=1 \)

Statement

The laws of indices are rules for simplifying expressions involving powers. For any non-zero number \(a\):

\[ a^m a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \frac{1}{a^n}, \quad a^0 = 1 \]

Why they’re true

  • Multiplication law: \(a^m\) means \(a\) multiplied by itself \(m\) times. Combining \(a^m \times a^n\) gives \(a^{m+n}\).
  • Division law: Cancelling common factors leads to \(a^m/a^n = a^{m-n}\).
  • Power of a power: Repeated multiplication gives \((a^m)^n = a^{mn}\).
  • Negative indices: \(a^{-n}\) means the reciprocal, \(1/a^n\).
  • Zero index: Dividing \(a^n\) by itself (\(a^n/a^n\)) gives \(a^0=1\).

Recipe (how to use them)

  1. If multiplying powers with the same base, add exponents.
  2. If dividing powers with the same base, subtract exponents.
  3. If raising a power to another power, multiply exponents.
  4. If the exponent is negative, rewrite as a reciprocal.
  5. If the exponent is zero, the value is 1 (for \(a \neq 0\)).

Spotting it

These laws apply when bases are the same, e.g., simplifying \(2^5 × 2^3\) or \((x^4)^2\).

Common pairings

  • Simplifying algebraic expressions with powers.
  • Standard form calculations.
  • Exponential growth/decay in science and finance.

Mini examples

  1. \(2^3 × 2^4 = 2^{3+4} = 2^7 = 128\).
  2. \(\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625\).
  3. \((3^2)^3 = 3^{2×3} = 3^6 = 729\).
  4. \(7^{-2} = 1/7^2 = 1/49\).
  5. \(10^0 = 1\).

Pitfalls

  • Bases must be the same: \(2^3 × 3^3 ≠ 6^3\).
  • Do not confuse signs: \(a^{-n} ≠ -a^n\).
  • Zero base caution: \(0^0\) is undefined.

Exam strategy

  • Always check that the base is the same before applying the laws.
  • Write each step clearly when adding, subtracting, or multiplying powers.
  • Convert negative or fractional indices into reciprocals/roots for clarity.

Summary

The laws of indices simplify working with powers: add for multiplication, subtract for division, multiply for powers of powers, flip for negatives, and zero gives 1. These are core rules for handling exponential expressions.