Please Explain How To Simplify?

Question

1. √80
2. 2√27
3. 2/ √10
4. √32 + √72

Answer 1

I would like to know if you are just looking for the answer or if you really want to learn. Anyway:

1. √80 is not a perfect square root. What number can multiply to get 80 and are perfect square roots (perfect square roots: 1=1*1, 4=2*2, 9=3*3...) 80/4=20, so √(4*20) = √80 --- 20/4=5, so √(4*4*5) = √80. Now take square roots of whatever you can. √4 = 2, so 2*2√5, and then 4√5

2. 2√27 9*3=27 so 2√(9*3) = 2√27. Square root what you can --- 2*3(square root of 9)√3(left (not perfect square) = 2√27 (2*3√3 = 2√27). Multiply --- 6√3

3. 2/ √10 --- square root can never be on the bottom. to get rid of it, you must multiply by √10/√10 because that equals to 1 and multiplying by one does not affect the problem. So --- 2/ √10 *√10/√10 -------- 2√10/10 (√10*√10=10). 2√10/10 can be simplified by dividing top and bottom by 2, so --- √10/5

4. √32 + √72 --- √32 = √(16*2) = 4√2
√72 = √(4*18) = √(4*(9*2)) = 2*3√2 = 6√2
4√2 + 6√2
Now ignore the √2's and add 4+6 to get 10
then add the √2 back to get 10√2

If you need more help, contact me, best answer?

COMMENT: It took me a long time to write this and i didnt realize that in that time someone was answering. Hopefully i explained more

Answer 2

A radical is not simplified if any of the factors of the radicand (that's the number under the radical sign) is a perfect square.
In #1, 80 can be written as the product of 16 and 5; 16 is a perfect square so take it's square root. 5 is not a perfect square, so you will have 4 sq rt 5

#2, 27 can be written as 3 * 9; 9 is a perfect square, so you have 2 * 3 sq rt 3, or 6 sq rt 3

#3 You can't have a radical in the denominator, so multiply both numerator and denominator by sq rt 10; this gives you 2 sqrt 10 in the numerator, and sq rt 10 times sq rt 10, which equals 10, in the denominator. Simplify by cancelling the 2, so your final answer is sq rt 10 / 5

#4 You can write sq rt 32 as 4 sr rt 2 (32 can be written as 16 * 2) and write sq rt 72 as 6 sq rt 2 ( 72 can be written as 36 * 2). You can add the "coefficients" outside the radicals, so you have 10 sq rt 2