Student Question

# How do you calculate the square root and cube root of a number?

The easiest way to find roots is through prime factorization. In order for a whole number to be a perfect square, each of its prime factors must have an even exponent. In other words, all of its distinct prime factors must occur in multiples of 2.

For example, 100 = 2^2*5^2 and 144=3^2*2^4

This also works backwards: if you multiply a bunch of prime numbers raised to even powers, you get a perfect square.

3^2*5^2*7^2=11025 ; \sqrt[11025]=105

To find the square root of a number using prime factorization, simply divide the exponent for each prime factor by two and then multiply what's left. We can see this in the previous three examples.

2*5=10

3*2^2=12

3*5*7=105

Finding a cube root follows much the same reasoning. In order to be a perfect cube, each of the prime factors of an integer must be raised to a power of 3.

216=3^3*2^3

To find a cube root, simply divide the exponent for each prime factor by 3.

3*2=6 and ^3\sqrt[216]=6

In cases where the number is not a perfect square (or cube), take the root of the prime factors that are raised to an even power (or multiple of 3) and leave the rest under the root sign.

\sqrt[12]=\sqrt[2^2]*\sqrt[3]=2\sqrt[3]