Laws of Indices (Core)

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.