# Eighth Root

LAST UPDATE: September 24th, 2020

## Definition – What is an eighth root?

The eighth root of a number is the number that would have to be multiplied by itself 8 times to get the original number.
For example, the eighth root of 6,561 is 3 as 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 is 6,561.
The eighth root of 57,536 is 4, as 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 is 57,536.

## Formula – How to calculate the eighth root

Even for perfect eighth root numbers, the eighth root can be difficult to calculate by hand.
The most basic techniques involve trial and error.
Example:
Find the eighth root of 16,777,216
1) Try a number – 5 : 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 390,625 (too low)
2) Try another number that is more than 5 – 6 : 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 = 1,679,616 (too low)
3) Try a number that is more than 6 – 10 – 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000 (too high)
4) Try a number in between 6 and 10 – 8 – 8 x 8 x 8 x 8 x 8 x 8 x 8 x 8 = 16,777,216 (answer)
An eighth root can also be calculated as the same number to the exponent 1/8.
8√number = number1/8