By Cam Merritt – Updated Aug 30, 2022
Exponentiation means multiplying a base number by itself a specified number of times. For example, 2³ equals 2×2×2 = 8. When the exponent is a fraction, the operation reverses: you are looking for a root of the base.
In mathematics, raising a number to a power is called exponentiation. An exponential expression has a base—the number being raised—and an exponent—the power. For instance, in 2³ the base is 2 and the exponent is 3. Raising a base to the second power is called squaring; raising it to the third power is called cubing. Exponents are normally written as superscripts (e.g., 2³) or with a caret notation (2^3) for devices that don’t support superscripts.
Roots are the inverse operation of exponents. If 2⁴ = 16, then the 4th root of 16 is 2. Similarly, 729 = 9³ and 9 is the 3rd root; 729 = 3⁶ and 3 is the 6th root. The 2nd root is known as the square root, and the 3rd root as the cube root.
When the exponent is a fraction, the denominator indicates the root you must take. For example, 125^(1/3) asks for the cube root of 125, which is 5 because 5×5×5 = 125. Likewise, 256^(1/4) seeks the 4th root of 256; 4×4×4×4 = 256, so the result is 4.
Fractional exponents with numerators greater than one combine a root with a power. In 8^(2/3), the denominator 3 tells you to take the cube root, while the numerator 2 instructs you to square the result. Whether you start by taking the cube root of 8 (which is 2) and then squaring it, or by squaring 8 (which is 64) and then taking its cube root, the outcome is the same: 4.
This “numerator as power, denominator as root” rule applies to all exponents, including whole numbers and fractions with a numerator of one. For example, 9² is equivalent to 9^(2/1). Raising 9 to the second power gives 81; the 1st root of 81 is 81 itself. Likewise, 9^(1/2) reduces to taking the square root of 9, yielding 3. The rule holds, but in these special cases one step can be omitted.
