Question

simplify (1/2-√3) +(5/3-√2)-(1/√2-√3)

Answer 1

(1/2+√3)+(2/√5-√3)+(1/2-√5)

(1/2+√3)+(2/√5-√3)+(1/2-√5)=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)

(1/2+√3)+(2/√5-√3)+(1/2-√5)=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)=(2-√3)/{2²-(√3)²}+2(√5+√3)/{(√5)²-(√3)²}+(2+√5)/{2²-(√5)²}

(1/2+√3)+(2/√5-√3)+(1/2-√5)=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)=(2-√3)/{2²-(√3)²}+2(√5+√3)/{(√5)²-(√3)²}+(2+√5)/{2²-(√5)²}=(2-√3)/(4-3)+2(√5+√3)/(5-3)+(2+√5)/(4-5)

(1/2+√3)+(2/√5-√3)+(1/2-√5)=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)=(2-√3)/{2²-(√3)²}+2(√5+√3)/{(√5)²-(√3)²}+(2+√5)/{2²-(√5)²}=(2-√3)/(4-3)+2(√5+√3)/(5-3)+(2+√5)/(4-5)=2-√3+{2(√5+√3)/2}-(2+√5)

(1/2+√3)+(2/√5-√3)+(1/2-√5)=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)=(2-√3)/{2²-(√3)²}+2(√5+√3)/{(√5)²-(√3)²}+(2+√5)/{2²-(√5)²}=(2-√3)/(4-3)+2(√5+√3)/(5-3)+(2+√5)/(4-5)=2-√3+{2(√5+√3)/2}-(2+√5)=2-√3+√5+√3-2-√5

(1/2+√3)+(2/√5-√3)+(1/2-√5)=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)=(2-√3)/{2²-(√3)²}+2(√5+√3)/{(√5)²-(√3)²}+(2+√5)/{2²-(√5)²}=(2-√3)/(4-3)+2(√5+√3)/(5-3)+(2+√5)/(4-5)=2-√3+{2(√5+√3)/2}-(2+√5)=2-√3+√5+√3-2-√5=0 (Proved)

Answer 2

-13+√3

This is the answer, I have not explained it properly because it is too long to write it on my mobile, so please cooperate with me and just try to find how it came. I know you will definitely find this answer very easy.

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