Question

Rationalize the denominator and simplify

1)  4 + root 5  +    4 - root 5
   _________       ________
    4 - root 5          4 + root 5

2)     3      +         2
   _______     ________
   5- root 3      5 + root 3

3)  root 5 + root 2  +  root 5 - root 2
    ____________      ____________
    root 5 - root 2        root 5 + root 2

Answer 1

1)(4+5^1/2)/(4-5^1/2)+(4-5^1/2)/(4+5^1/2)=(4+5^1/2)(4+5^1/2)+(4-5^1/2)(4-5^1/2)/4^2-5^1/2*2=2(4^2+5^1/2*2)/16-5=2(21)/11=42/11
2)3/(5-3^1/2)+2/(5+3^1/2)=[3(5+3^1/2)+2(5-3^1/2)]/(5^2-3^1/2*2)=(15+3*3^1/2+10-2*3^1/2)/16=(25+3^1/3)/16
3)(5^1/2+2^1/2)/(5^1/2-2^1/2)+(5^1/2-2^1/2)/(5^1/2+2^1/2)
 =(5^1/2+2^1/2)^2+(5^1/2-2^1/2)^2/(5^1/2*2-2^1/2*2)
 =2(5^1/2*2+2^1/2*2)/3
 =2(7)/3=14/3

Answer 2

1. (4+5^1/2)/(4-5^1/2)*(4+5^1/2)/(4+5^1/2)+(4-5^1/2)/(4+5^1/2)*(4-5^1/2)*(4-5^1/2)
= (16+5+8*5^1/2)/(16-5)+(16+5-8*5^1/2)/(16-5)
= (21+8*5^1/2+21-8*5^1/2)/11
=42/11