Question

1+√2/√5+√3 + 1-√2/√5-√3 (WITH DETAILED STEPS)

Answer 1

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


first of all rationalise , both terms
( 1 + √2)/( √5 + √3)

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

= ( √5 +√10-√3 -√6 )/( 5 -3)

= ( √5 + √10-√3 -√6)/2

similarly,

rationalise of (1 -√2)/( √5 -√3)

= (1-√2)(√5 +√3)/( √5 -√3)(√5 + √3)

= ( √5 + √3 -√10 - √6)/(5 -3)

= ( √5 + √3 -√10 -√6)/2

add , both ,

( √5 + √10 -√3 -√6)/2 + (√5 +√3 -√10-√6)/2

= ( 2√5 -2√6)/2

= √5 - √6

Answer 2

(1+√2)/(√5+√3) + (1-√2)/(√5 -√3)

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

= [√5 -√3 +√10 -√6 +√5+√3 -√10 -√6]/ [ (√5)² - (√3)²]

=[2√5-2√6]/[5-3]

= 2[√5 -√6] /2
= √5 -√6