Exponents

Fractional Exponents

Raising a number to a fraction is the same as taking the demoninator corresponding root of the number. For example \( 9^{\frac{1}{2}} = \sqrt{9} \) and \( 4^{\frac{1}{3}} = \sqrt[3]{4} \).

The reason that fractional exponents evaluate to the root of a number can be explained by reference to the rules of exponents. We know that when you multiple exponents they “add”, such as \( 2^5 \times 2^2 = 2^7 \). So when we multiply \( x^{\frac{1}{2}} \times x^{\frac{1}{2}} \) we know that the two halves must add together to equal 1 in the form \( x^1 \) which is the same as just \( x \). So we multiplied together two of the same things to get another thing which is essentially the definition of a square root, i.e. \( \sqrt{x} \times \sqrt{x} = x \). Therefore raising \( x^{\frac{1}{2}} \) must be equal to \( \sqrt{x} \).

Raising a number to a fraction with a numerator greater than one means that you first root the number based on the demoninator and then raise it to the power of the numerator. For example: \( 2^{\frac{2}{3}} = \sqrt[3]{2}^{2} = 2\sqrt{2} \).