Definition Of Nth Root Of A Real Number



nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x:

r^n=x, where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc.

For Example:

  • 2 is a square root of 4, since 2^2 = 4.
  • −2 is also a square root of 4, since (−2)^2 = 9.

Any non-zero number, considered as complex number, has n different 'complex roots of degree n' (n-th roots), including those with zero imaginary part, i.e. any real roots. The root of 0 is zero for all degrees n, since 0n = 0. In particular, if n is even and x is a positive real number, one of its n-th roots is positive, one is negative, and the rest (when n > 2) are complex but not real; if n is even and x is a negative real, none of the n-th roots is real. If n is odd and x is real, one n-th root is real and has the same sign as x, while the other (n−1) roots are not real. Finally, if x is not real, then none of its n-th roots is real.


Multiplication and Division

Multiplications under the root sign looks like this:

(If n is even, a and b must both be ≥ 0)

This can help us simplify equations in algebra, and also make some calculations easier:


It also works for division:

(a≥0 and b>0)
(b cannot be zero, as we can't divide by zero)