Step by step
The question
Question: simplify the 4th root of 162.
The small 4 says how many times.
The 4th root of 162 asks: which number, times itself 4 times, makes 162?
1 Find the biggest 4th power
4th powers are a number times itself 4 times.
2 × 2 × 2 × 2 is 16.
3 × 3 × 3 × 3 is 81.
We try the biggest first.
Does 81 go into 162?
Yes: 162 divided by 81 is 2.
Careful!
9 is a square, not a 4th power.
The 4th root of 9 is not 3.
We need a number times itself 4 times.
2 Split the root
So 162 is 81 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 81 times the 4th root of 2.
The 4th root of 81 is 3, because 3 × 3 × 3 × 3 is 81.
We write 3 in place of the 4th root of 81.
We leave out the times sign.
2 has no 4th power in it, so we are done.
So the 4th root of 162 = 3 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.
3 × 3 × 3 × 3 is 81.
Times 2 is 162.
We get 162 back.
So the answer is right!
Let's look back.
First: find the biggest 4th power.
Then: split the root.
Last: check.