Alicia H. answered • 05/01/19
Creative & Resourceful Math Tutor
You want to simplify every radical individually first. Then you can combine the coefficients of radicals only if they have the same radicand (the same values under the radical symbol).
To break down a radical in simplest form:
- You will need two radicals
- In the first radical, put the biggest perfect square that divides into the number under the radical (called the radicand) --> The biggest perfect square that divides into 75 is 25 so 25 will go under your first radical.
- Recall that perfect squares are formed by multiplying a number by itself (ex: 1, 4, 9, 16, 25, ... are perfect squares because you multiply 1 by 1 to get 1, 2 by 2 to get 4, and so on)
- In the second radical, put the value leftover when you divide the original value (75) by the biggest perfect square factor (25). So you will put 3 under the second radical since 75 ÷ 25 = 3
- Simplify the radical with with perfect square under it. Remember, after you simplify a radical, it is no longer under a radical symbol. This means √75 simplifies to √25·√3 which ultimately becomes 5√3
When you simplify the radicals, you should get 5√3 - 4√5 + 3√3 + 9√5
Those coefficients (the numbers in front of the radical symbol) are being multiplied to the radical. After I simplified square root of 20 to 2√5, I had to multiply the 2 that I just got from simplifying by the 2 that was originally in front of the radical. That's where the new coefficient of 4 comes from. Similarly, √45 = √9·√5 = 3√5, so I need to multiply this 3 out front by the original coefficient 3 to get 9√5.
Now that my radicals are simplified, I only combine the coefficients (not the radicands!) on terms with the same radicand. Think of it like adding like terms - you wouldn't combine 2x + 3y, and you wouldn't say 2x + 2x is 4x2 either. You will combine the coefficients on 5√3 and 3√3 and combine the coefficients on -4√5 and 9√5.
In simplest radical form, you get 8√3 + 5√5.
