Rational-Irrational Numbers-9

Simplify each of the following by rationalizing the denominator:

(i) (√5 - √2) / (√5 + √2) (ii) (2 √3 + 3 √2) / (2 √3 - 3 √2)

(iii) ( √2 + √7) / ((5 √7 + 7 √2)

Solution: (√5 - √2) / (√5 + √2) = (√5 - √2) / (√5 + √2) x (√5 - √2) / (√5 - √2)

= (5 + 2 - 2 √5 x √2) / [ (√5)² - (√2)²] = (5 + 2 - 2 √10) / (5 - 2) = (7 - 2 √10) / 3

(ii) (2 √3 + 3 √2) / (2 √3 - 3 √2) = (2 √3 + 3 √2) / (2 √3 - 3 √2) x (2 √3 + 3 √2) / (2 √3 + 3 √2)

= (4 x 3 + 9 x 2 + 2 x 2 √3 x 3 √2) / [(2 √3)² - (3 √2)²]

= (12 + 18 + 12 √6) / (12 - 18) = (30 + 12 √6) / (-6) = - (5 + 2 √6)

(iii) ( √2 + √7) / ((5 √7 + 7 √2) = ( √2 + √7) / ((5 √7 + 7 √2) x ((5 √7 - 7 √2) / ((5 √7 - 7 √2)

= [ ( 5 √14 - 7 x 2 + 5 x 7 - 7 √14 ) / (5 √7)² - (7 √2)² ]

= (21 - 2 √14) / (175 - 98) = (21 - 2√14) / 77