find the value of a and b if ( root2 + root3) / (3root3 + 2root3) =a-b root6 - Maths - Polynomials

Question

find the value of a and b if ( root2 + root3) / (3root3 +
2root3) =a-b root6

Ajanta Trivedi · Accepted Answer

if the given equation is: $\frac{\sqrt{2} + \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} = a - b \sqrt{6}$

rationalising the denominator from the LHS part

$\frac{\sqrt{2} + \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} = \frac{\sqrt{2} + \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} * \frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}} = \frac{\sqrt{2} (3 \sqrt{2} - 2 \sqrt{3}) + \sqrt{3} (3 \sqrt{2} - 2 \sqrt{3})}{(3 \sqrt{2})^{2} - (2 \sqrt{3})^{2}} = \frac{6 - 2 \sqrt{6} + 3 \sqrt{6} - 6}{18 - 12} = \frac{\sqrt{6}}{6}$

therefore

$\frac{\sqrt{6}}{6} = a - b \sqrt{6}$

comparing the rational and irrational parts from the both sides of the equation:

$a = 0 a n d b = - \frac{1}{6}$

hope this helps you.

cheers!!