### Product Rule - **Definition:** When multiplying two powers with the same base, add the exponents. - **Formula:** $$a^m \cdot a^n = a^{m+n}$$ - **Example:** $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$ ### Quotient Rule - **Definition:** When dividing two powers with the same base, subtract the exponents. - **Formula:** $$\frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0$$ - **Example:** $$\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$$ ### Power Rule - **Definition:** When raising a power to another power, multiply the exponents. - **Formula:** $$(a^m)^n = a^{mn}$$ - **Example:** $( (4^2)^3 = 4^{2 \cdot 3} = 4^6 = 4096 )$ ### Zero Exponent Rule - **Definition:** Any non-zero number raised to the power of zero is 1. - **Formula:** $$a^0 = 1, \quad a \neq 0$$ - **Example:** $5^0 = 1$, $(-7)^0 = 1$ ### Negative Exponent Rule - **Definition:** A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. - **Formula:** $$a^{-n} = \frac{1}{a^n}, \quad a \neq 0$$ - **Example:** $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ ### Product to a Power Rule - **Definition:** When a product is raised to a power, each factor is raised to that power. - **Formula:** $$(ab)^n = a^n b^n$$ - **Example:** $( (2x)^3 = 2^3 x^3 = 8x^3 )$ ### Quotient to a Power Rule - **Definition:** When a quotient is raised to a power, both the numerator and the denominator are raised to that power. - **Formula:** $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \quad b \neq 0$$ - **Example:** $$\left(\frac{x}{y}\right)^2 = \frac{x^2}{y^2}$$ ### Fractional Exponents - **Definition:** A base raised to a fractional exponent is equivalent to taking the nth root of the base raised to the mth power. - **Formula:** $$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$ - **Example:** $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$
