# Tenth Root

LAST UPDATE: September 24th, 2020

## Definition – What is a tenth root?

The tenth root of a number is the number that would have to be multiplied by itself 10 times to get the original number.
For example, the tenth root of 59,049 is 3 as 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 is 59,049.
The ninth root of 690,432 is 4, as 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 is 690,432.

## Formula – How to calculate the tenth root

Even for perfect tenth root numbers, the tenth root can be difficult to calculate by hand.
The most basic techniques involve trial and error.
Example:
Find the tenth root of 1,073,741,824
1) Try a number – 5 : 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 9,765,625 (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 x 6 x 6 = 60,466,176 (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 x 10 x 10 = 10,000,000,000 (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 x 8 = 1,073,741,824 (answer)
An ninth root can also be calculated as the same number to the exponent 1/10.
10√number = number1/10