# Sixth Root

LAST UPDATE: September 24th, 2020

## Definition – What is a sixth root?

The sixth root of a number is the number that would have to be multiplied by itself 6 times to get the original number.
For example, the sixth root of 729 is 3 as 3 x 3 x 3 x 3 x 3 x 3 is 729.
The sixth root of 4,096 is 4, as 4 x 4 x 4 x 4 x 4 x 4 is 4,096.

## Formula – How to calculate the sixth root

Even for perfect sixth root numbers, the sixth root can be difficult to calculate by hand.
The most basic techniques involve trial and error.
Example:
Find the sixth root of 262,144
1) Try a number – 5 : 5 x 5 x 5 x 5 x 5 x 5 = 163,840 (too low)
2) Try another number that is more than 5 – 6 : 6 x 6 x 6 x 6 x 6 x 6 = 46,656(too low)
3) Try a number that is more than 6 – 10 – 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 (too high)
4) Try a number in between 6 and 10 – 8 – 8 x 8 x 8 x 8 x 8 x 8 = 262,144 (answer)
A sixth root can also be calculated as the same number to the exponent 1/6.
6√number = number1/6