nth Root

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 ⁴√3⁴ = 3.

So 4 is the answer.

The even number root cannot be (-).
So ⁴√   cannot be (-).

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 (-).

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.

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 ⁴√2⁴ (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 ⁴√16x¹²y⁸.
The exponent of x is 12: even.
So x can be (-).

If x is (-),

the given, ⁴√16x¹²y⁸, 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.