Laws of Exponents

Math Exponent Rules

Math Exponent Rules

Rule Mathematical Expression Example
Product of Powers \[ a^m \times a^n = a^{m+n} \] \[ 2^1 \times 2^2 = 2^3 \]
Power of a Product \[ (a \times b)^m = a^m \times b^m \] \[ (2 \times 3)^2 = 2^2 \times 3^2 \]
Quotient of Powers \[ \frac{a^m}{a^n} = a^{m-n} \] \[ \frac{3^2}{3^1} = 3^{2-1} = 3^1 \]
Power of a Quotient \[ \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \] \[ \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} \]
Negative Exponent \[ a^{-m} = \frac{1}{a^m} \] \[ 2^{-1} = \frac{1}{2^1} \]
Zero Exponent \[ a^0 = 1 \] \[ 3^0 = 1 \]
Power of a Power \[ (a^m)^n = a^{m \times n} \] \[ (2^1)^2 = 2^{1 \times 2} = 2^2 \]
Fractional Exponent \[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \] \[ 8^{\frac{2}{3}} = \sqrt[3]{8^2} = 4 \]

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