Step by step
The question
Question: simplify the 4th root of 32.
The small 4 says how many times.
The 4th root of 32 asks: which number, times itself 4 times, makes 32?
1 Find the biggest 4th power
4th powers are a number times itself 4 times.
2 × 2 × 2 × 2 is 16.
We try the biggest first.
Does 16 go into 32?
Yes: 32 divided by 16 is 2.
Careful!
4 is a square, not a 4th power.
The 4th root of 4 is not 2.
We need a number times itself 4 times.
2 Split the root
So 32 is 16 times 2.
We write it under the root.
Rule: a root over a times can be split into two roots.
So it is the 4th root of 16 times the 4th root of 2.
The 4th root of 16 is 2, because 2 × 2 × 2 × 2 is 16.
We write 2 in place of the 4th root of 16.
We leave out the times sign.
2 has no 4th power in it, so we are done.
So the 4th root of 32 = 2 times the 4th root of 2.
3 Check
Let's check: we take the answer to the power 4.
The 4th root of 2 to the power 4 is just 2.
2 × 2 × 2 × 2 is 16.
Times 2 is 32.
We get 32 back.
So the answer is right!
Let's look back.
First: find the biggest 4th power.
Then: split the root.
Last: check.