Laws of indices

The Laws of Indices (or Exponent Rules) are mathematical rules for simplifying expressions involving powers.

Key laws include:

Multiplication Law

\[ a^m \times a^n = a^{m+n} \] When multiplying powers with the same base, add the exponents.

Division Law

\[ \frac{a^m}{a^n} = a^{m-n} \] When dividing powers with the same base, subtract the exponents.

Power of a Power Law

\[ (a^m)^n = a^{m \times n} \] When raising a power to another power, multiply the exponents.

Power of a Product Law

\[ (ab)^n = a^n \times b^n \] When raising a product to a power, apply the exponent to each factor.

Power of a Quotient Law

\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \] When raising a quotient to a power, apply the exponent to both numerator and denominator.

Zero Exponent Law

\[ a^0 = 1 \] Any non-zero number raised to the power of zero equals 1.

Negative Exponent Law

\[ a^{-n} = \frac{1}{a^n} \] A negative exponent indicates the reciprocal of the base raised to the positive exponent.