Question

simplify 1/(1+square root of 2) + 1/( square root of 2 + square root of 3) + 1/(square root of 3 + square root of 4)

Answer 1

1/(1+√2) + 1/(√2 + √3) + 1/(√3 + √4)

rationalize each term:
1/(1+√2)
= 1/(1+√2) * (1-√2)/(1-√2)
= (1-√2)/-1
= √2 - 1

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

similarly
1/(√3 + √4)
= -√3 + √4

then:
1/(1+√2) + 1/(√2 + √3) + 1/(√3 + √4)
= √2 - 1 + (-√2 + √3) + (-√3 + 2)
= 1