# nth Root

How to solve the nth-root of a number: 4 examples and their solutions.

## Example 1

### Example

### Solution

Solving an nth root is similar to

solving a square root or a cube root.

To solve an nth root,

change the number inside the radical sign

to the power of n.

The given radical is a 4th root.

So change 81 to the power of 4:

81 = 3^{4}.

Cancel the 4th root and the exponent 4.

Then ^{4}√3^{4} = 3.

So 4 is the answer.

The even number root cannot be (-).

So ^{4}√ cannot be (-).

## Example 2

### Example

### Solution

The given radical is a 5th root.

So change -32 to the power of 5:

-32 = (-2)^{5}.

Cancel the 5th root and the exponent 5.

Then ^{5}√(-2)^{5} = -2.

So -2 is the answer.

The odd number root can be (-).

So, just like ^{5}√-32 = -2,^{5}√ can be (-).

## Example 3

### Example

### Solution

Cancel the square root and the square.

Then √x^{2} = x.

When solving a radical with a variable

that can be (-),

think of the cases

when the signs of the given and the result

are different.

See the given radical √x^{2}.

x is squared.

So x can be (-).

If x is (-),

the given, √x^{2}, is still (+).

(The given is an even root number.)

But the result x is (-).

The signs of the given and the result

are different.

Then, to make the signs the same,

write the absolute value sign

to the result x.

So |x| is the answer.

## Example 4

### Example

### Solution

The given radical is a 4th root.

So change the factors to the powers of 4.

16 = 2^{4}

x^{12} = (x^{3})^{4}

y^{8} = (y^{2})^{4}

Power of a Power

Cancel the 4th root and the exponents 4.

Then ^{4}√2^{4} (x^{3})^{4} (y^{2})^{4}

= 2 x^{3} y^{2}.

When solving a radical with a variable,

think of the cases

when the signs of both sides are different.

See the given radical ^{4}√16x^{12}y^{8}.

The exponent of x is 12: even.

So x can be (-).

If x is (-),

the given, ^{4}√16x^{12}y^{8}, is still (+).

(The given is an even root number.)

But the result, 2x^{3}y^{2}, is (-).

The signs of the given and the result

are different.

Then, to make the signs the same,

write the absolute value sign

to the x^{3}.

So 2|x^{3}|y^{2} is the answer.