Nth root
Logarithmic calculation

The principal n th root of a positive number can be computed using logarithms. Starting from the equation that defines r as an n th root of x, namely r n = x, {\displaystyle r^{n}=x,} with x positive and therefore its principal root r also positive, one takes logarithms of both sides (any base of the logarithm will do) to obtain

n log b r = log b x hence log b r = log b x n. {\displaystyle n\log _{b}r=\log _{b}x\quad \quad {\text{hence}}\quad \quad \log _{b}r={\frac {\log _{b}x}{n}}.}

The root r is recovered from this by taking the antilog:

r = b 1 n log b x. {\displaystyle r=b^{{\frac {1}{n}}\log _{b}x}.}

(Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.)

For the case in which x is negative and n is odd, there is one real root r which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain | r | n = | x |, {\displaystyle |r|^{n}=|x|,} then proceeding as before to find | r |, and using r = −| r |.
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