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 = 34.

Cancel the 4th root and the exponent 4.

Then 434 = 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 √x2 = 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 √x2.
x is squared.
So x can be (-).

If x is (-),

the given, √x2, 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 = 24
x12 = (x3)4
y8 = (y2)4

Power of a Power

Cancel the 4th root and the exponents 4.

Then 424 (x3)4 (y2)4
= 2 x3 y2.

When solving a radical with a variable,
think of the cases
when the signs of both sides are different.

See the given radical 416x12y8.
The exponent of x is 12: even.
So x can be (-).

If x is (-),

the given, 416x12y8, is still (+).
(The given is an even root number.)

But the result, 2x3y2, 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 x3.

So 2|x3|y2 is the answer.