Step by step
The question
Question: simplify √45.
√45
1 Find the biggest square number
Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.
We list them up to half of 45.
A bigger one can't go in.
Square numbers: 4, 9, 16
We try the biggest first.
Does 16 go into 45?
No.
Does 9 go into 45?
Yes: 45 divided by 9 is 5.
45 ÷ 16 ✗ 45 ÷ 9 = 5 ✓
2 Split the root
So 45 is 9 times 5.
We write it under the root.
= √9 × 5
Rule: a root over a times can be split into two roots.
√(a × b) = √a × √b
So √(9 × 5) is √9 times √5.
= √9 × √5
√9 is 3, because 3 times 3 is 9.
3 × 3 = 9
We write 3 in place of √9.
= 3 × √5
We leave out the times sign: 3√5 means 3 times √5.
5 has no square number in it, so we are done.
So √45 = 3√5.
= 3√5
3 Check
Let's check: we square the answer.
We must get 45 back.
3 × 3, and √5 × √5 = 5.
Check: (3√5)2 = 3 × 3 × 5 = 45
We get 45.
So 3√5 is right!
Let's look back.
First: find the biggest square number.
Then: split the root.
Last: check.
Answer
= 3√5