Properties of Exponents

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

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

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

When raising a product to a power, apply the exponent to each factor: $$(ab)^n = a^n b^n$$ .

When raising a quotient to a power, apply the exponent to both the numerator and the denominator: $$\frac{a}{b}^n = \frac{a^n}{b^n}$$ .

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

Any non-zero base raised to the zero power equals one: $$a^0 = 1$$ , where a ≠ 0.

$$\frac{4x^7y^3}{3}$$ .

$$\frac{1}{x^{11}}$$ .

$$\frac{x^6}{2z^4}$$ .