# Ninth Root

LAST UPDATE: September 24th, 2020

## Definition – What is a ninth root?

The ninth root of a number is the number that would have to be multiplied by itself 9 times to get the original number.
For example, the ninth root of 19,683 is 3 as 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 is 19,683.
The ninth root of 230,144 is 4, as 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 is 230,144.

## Formula – How to calculate the ninth root

Even for perfect ninth root numbers, the ninth root can be difficult to calculate by hand.
The most basic techniques involve trial and error.
Example:
Find the ninth root of 134,217,728
1) Try a number – 5 : 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 1,953,125 (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 = 10,077,696 (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 = 1,000,000,000 (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 = 134,217,728 (answer)
An ninth root can also be calculated as the same number to the exponent 1/9.
9√number = number1/9