nth Root

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.

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 481

Simplify the given expression. Fourth root [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 (+).

So 3 is the answer.

Example 2: Simplify 5-32

Simplify the given expression. Fifth root [-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.

So -2 is the answer.

Example 3: Simplify √(-7)2

Simplify the given expression. Square root [(-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

So 7 is the answer.

Example 4: Simplify √-72

Simplify the given expression. Square root [-7^2]

For an even root,
if the radicand is (-),
then that radical is not a real number.
(Radicand: the number inside the radical sign)

The radicand of √-72 is -72: (-).

So √-72 is not a real number.

Example 5: Simplify √x2

Simplify the given expression. Square root [x^2]

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|.

So |x| is the answer.

Example 6: Simplify √x4

Simplify the given expression. Square root [x^4]

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.

So x2 is the answer.

Example 7: Simplify 5x15

Simplify the given expression. Fifth root [x^15]

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.

So x3 is the answer.

Example 8: Simplify 416x12y8

Simplify the given expression. Fourth root [16 x^12 y^8]

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.

So 2|x3|y2 is the answer.

Example 9: Simplify 327x12y15

Simplify the given expression. Cube root [27 (x - 8)^12 y^15]

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.