Radical Simplification

[Abdul Quadeer]

1. The problem statement, all variables and given/known data
I got this expression while solving a problem.

sin(theta)=[√3+1-√(4-2√3)]/4
Using calculator I got RHS=0.5
How do we simplify it without using a calc?


2. Relevant equations



3. The attempt at a solution

[ehild]

Try to write 4-2√3 as the square of (√3-a) . What is a?

ehild

[HallsofIvy]

ehild, did you intend that to be \sqrt{(3- a)} or (\sqrt{3})- a?

[ehild]

Naturally, √3 +a = 1.7320508.. +a. You have to perform a higher operation first, and square root is of higher order than addition or subtraction. So is multiplication to addition. 2*3+4 =10 and it is not the same as 2*(3+4)=14. Or 6/2+3=6, and not 6/(2+3)=1.2. √9+16=19 and it is not √(9+16)=5. The square of (√3-1) is (√3-1)^2 = 4-2√3 and not √3-1^2=√3-1. Or it is on the other way at other parts of the world?

ehild

[Mentallic]

Naturally, √3 +a = 1.7320508.. +a. You have to perform a higher operation first, and square root is of higher order than addition or subtraction. So is multiplication to addition. 2*3+4 =10 and it is not the same as 2*(3+4)=14. Or 6/2+3=6, and not 6/(2+3)=1.2. √9+16=19 and it is not √(9+16)=5. The square of (√3-1) is (√3-1)^2 = 4-2√3 and not √3-1^2=√3-1. Or it is on the other way at other parts of the world?

ehild

Yes?

Abdul Quadeer, a+b=\sqrt{(a+b)^2}, see if you can work some magic with this idea.

[Abdul Quadeer]

Thanks, I got the answer.
Mentallic- there is a slight error in your idea.
LHS =|a+b|

[Mentallic]

Haha yep :smile: I didn't want to complicate things by saying in the case that a+b>0 or anything like that. You know, the KISS principle. But I'm glad you were able to spot my simplification (in other words, the error, but I'm not willing to admit it :wink:)

[Abdul Quadeer]

:smile: