# Root Calculator (nth Root)

LAST UPDATE: September 24th, 2020

## Definition – What is a root?

A root is a number that is multiplied by itself the root number of times.

For example, the square root (root 2) of 16 (√16) is 4, as 42 (4 x 4) = 16.

The cubed root (root 3) of 27 (3√27) is 3, as 33 (3 x 3 x 3) = 27.

The 5th root of 1,024 (5√1024) is 4, as 45 (4 x 4 x 4 x 4 x 4) = 1,204.

The 2.5th root of 70 (2.5√70) is 5.47065, as 5.470652.5 = 70.

Square root, cubed root, 4th root, and any root are the most common examples of an nth root. Roots can also include decimal numbers (root 6.4, for example).

## Formula – How to calculate a root

Even for perfect root numbers, a root can be difficult to calculate by hand. The most basic techniques involve trial and error.

Example – Find the fourth root of 4,096 using trial and error:

• Try a number – 5 : 5 x 5 x 5 x 5 = 625 (too low)
• Try another number that is more than 5 – 6 : 6 x 6 x 6 x 6 = 1,296 (too low)
• Try a number that is more than 6 – 10 – 10 x 10 x 10 x 10 = 10,000 (too high)
• Try a number in between 6 and 10 – 8 – 8 x 8 x 8 x 8 = 4,096 (answer)

Example 2 – Find the fifth root of 32,768 using trial and error:

• Try a number – 5 : 5 x 5 x 5 x 5 x 5 = 3,125 (too low)
• Try another number that is more than 5 – 6 : 6 x 6 x 6 x 6 x 6 = 7,776(too low)
• Try a number that is more than 6 – 10 – 10 x 10 x 10 x 10 x 10 = 100,000 (too high)
• Try a number in between 6 and 10 – 8 – 8 x 8 x 8 x 8 x 8 = 32,768 (answer)