nth Root of a Number

nth Root of a Number

How to find the nth root of a number: examples and their solutions.

Example 1

Simplify the given expressions. 1: square root 49. 2: cube root 125. 3: fourth root 81. 4: fifth root 32.

To solve the nth root,
change the radicand to nth power form.

Then the nth root and the nth power are cancelled.
And the base can get out from the radical sign.

Simplifying a radical (part 1)

Example 2

Simplify the given expressions. 1: square root -49. 2: cube root -125. 3: fourth root -81. 4: fifth root -32.

1., 3.: The radicand of an even number root cannot be (-).
So it cannot be a real number.

There are no x values that satisfy
x2 = -49 or x4 = -81.

2., 4.: The radicand of an odd number root can be (-).
So it can be simplified just like (+).
(-5)3 = -125, (-2)5 = -32.

Example 3

Simplify the given expressions. 1: square root x^2. 2: square root x^4. 3: square root x^6. 4: fourth root (16(x^12)(y^8)).

Simplifying an even number root with x:

The signs of both 'given' and 'answer'
should be the same,
even if x is (-).

x2 = x

Put x = -1:
(-1)2 ≠ -1
(1 ≠ -1)

So add an absolute value sign
to make both sides equal:
x2 = |x|.

x4 = x2

Put x = -1:
(-1)4 = (-1)2
(1 = 1)

So you don't have to add any absolute value signs:
x2 is the answer.

x6 = x3

Put x = -1:
(-1)6 ≠ (-1)3
(1 ≠ -1)

So |x3| is the answer.

416x12y8 = 2⋅x3y2

Put x = y = -1:

416⋅(-1)12⋅(-1)8 ≠ 2⋅(-1)3⋅(-1)2
(1 ≠ -1)

x3 makes (-).
So 2|x3|y2 is the answer.

Example 4

Simplify the given expressions. 1: cube root x^3. 2: cube root x^6. 3: cube root x^9. 4: cube root (27((x - 8)^12)(y^15)).

Simplifying an odd number root with x:

You don't need to think of the signs
of the 'given' and the 'answer',
because the radicand of the odd number root
can be (-).