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.