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

⁴√81 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.

⁴√81 is an even root.

So check if
the signs of ⁴√81 and 3 are the same.

⁴√81 is (+).

3 is (+).

So 3 is the answer.

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.

So -2 is the answer.

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

So 7 is the answer.

Example 4: Simplify √-7²

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

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

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 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.

⁵√x¹⁵ 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 ⁴√16x¹²y⁸

⁴√16x¹²y⁸ 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⋅x³⋅y².

⁴√16x¹²y⁸ is an even root.

So check if
the signs of ⁴√16x¹²y⁸ and 2⋅x³⋅y²
are always the same.

1. If x is (-),

⁴√16x¹²y⁸ is (+).

2⋅x³⋅y² is,
2⋅(-)³⋅y² = (+)⋅(-)⋅(+),
(-).

So change x³ to |x³|.

2. If y is (-),

⁴√16x¹²y⁸ is (+).

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

So just leave y².

So 2|x³|y² is the answer.

Example 9: Simplify ³√27x¹²y¹⁵

³√27x¹²y¹⁵ 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)⁴⋅y⁵.

³√27x¹²y¹⁵ 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.

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