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 4√34 = 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 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.