In each of the following determine rational numbers a and b. (i) 5 + 3√3/7 + 4√3 = a + b √3 (ii) √11 - √7/√11 + √7 = a - b √77

Question

In each of the following determine rational numbers a and b.

(i) \(\frac{5+3\sqrt3}{7+4\sqrt3}\) = a + b√3

(ii) \(\frac{\sqrt11-\sqrt7}{\sqrt11+\sqrt7}\) = a - b√77

(iii) \(\frac{4+3\sqrt5}{4-3\sqrt5}\) = a + b√5

Answer 1

(i) \(\frac{5+3\sqrt3}{7+4\sqrt3}\) = a + b√3 given,

Rationalising factor for denominator is 7 - 4√3

On equating rational and irrational parts we get,

a = -1 and b = 1

(ii) \(\frac{\sqrt11-\sqrt7}{\sqrt11+\sqrt7}\) = a - b√77 given,

On equating rational and irrational parts we get,

a = \(\frac{9}{2}\), b = \(\frac{1}{2}\)

(iii) \(\frac{4+3\sqrt5}{4-3\sqrt5}\) = a + b√5 given,

On equating rational and irrational parts we have,

 a = \(\frac{-61}{29}\) and b = \(\frac{-24}{29}\)