 # nth Root How to find the nth root of a number: examples and their solutions.

## How to Solve To solve an nth root,
make the power of n,
then cancel the nth root and the power of n.

For an even root (n = 2, 4, 6, ...),
if the signs of both sides are not the same,
use the absolute value signs
to make the signs of both sides the same.

Absolute value

## Example 1: Simplify 4√81 481 is a fourth root.

So make 81 the power of fourth.

81 = 34

Cancel the fourth root and the power of fourth.

Then (given) = 3.

481 is an even root.

So check if
the signs of 481 and 3 are the same.

481 is (+).

3 is (+).

## Example 2: Simplify 5√-32 5-32 is a fifth root.

So make -32 the power of fifth.

-32 = (-2)5

Cancel the fifth root and the power of fifth.

Then (given) = -2.

5-32 is not an even root.

So you don't have to check the signs of both sides.

## Example 3: Simplify √(-7)2 Cancel the square and the square.

Then (given) = -7.

(-7)2 is an even root.

So check if
the signs of √(-7)2 and -7 are the same.

(-7)2 is (+).

-7 is (-).

So, to make -7 (+),
change -7 to |-7|.

|-7| = 7

## Example 4: Simplify √-72 For an even root,
then that radical is not a real number.

The radicand of √-72 is -72: (-).

So √-72 is not a real number.

## Example 5: Simplify √x2 Cancel the square and the square.

Then (given) = x.

x2 is an even root.

So check if
the signs of √x2 and x are always the same.

In most cases,
when the variable is (-),
both sides can have different signs.

So check for the case
when the variable is (-).

If x is (-),

x2 is (+).
x (brown) is (-).

The signs should be the same.

So, to make x (+),
change x to |x|.

## Example 6: Simplify √x4 To cancel the square root,
make a square.

x4 = (x2)2

Power of a power

Cancel the square and the square.

Then (given) = x2.

x4 is an even root.

So check if
the signs of √x4 and x2 are always the same
when the variable is (-).

If x is (-),

x4 is (+).
x2 is (+).

The signs are the same.

## Example 7: Simplify 5√x15 To cancel the cube root,
make a cube.

x15 = (x3)5

Cancel the fifth root and the power of fifth.

Then (given) = x3.

5x15 is not an even root.

So you don't have to check the signs of both sides.

## Example 8: Simplify 4√16x12y8 416x12y8 is a fourth root.

So change the radicand to the powers of fourth.

16 = 24
x12 = (x3)4
y8 = (y2)4

Cancel the fourth root and the powers of fourth.

Then (given) = 2⋅x3y2.

416x12y8 is an even root.

So check if
the signs of 416x12y8 and 2⋅x3y2
are always the same.

1. If x is (-),

416x12y8 is (+).

2⋅x3y2 is,
2⋅(-)3y2 = (+)⋅(-)⋅(+),
(-).

So change x3 to |x3|.

2. If y is (-),

416x12y8 is (+).

2⋅|x3|⋅y2 is,
2⋅|x3|⋅(-)2 = (+)⋅(+)⋅(+),
(+).

So just leave y2.

## Example 9: Simplify 3√27x12y15 327x12y15 is a cube root.

So change the radicand to the cubes.

27 = 33
(x - 8)12 = ((x - 8)4)3
y15 = (y5)3

Cancel the cube root and the cubes.

Then (given) = 3⋅(x - 8)4y5.

327x12y15 is not an even root.

So you don't have to check the signs of both sides.

So 3(x - 8)4y5 is the answer.