Question

(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = a - b * sqrt(6) . Find a and b.​

Answer 1

Given Equation :

$\sf { \dfrac{ \sqrt{2} + \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} } = a - b \sqrt{6} }$

To calculate :

Value of a and b.

Clarification :

In order to calculate the value of a and b, firstly we need to rationalize the denominator in L.H.S. After that, by comparing L.H.S and R.H.S, we'll finr the value of a and b.

How to rationalize the denominator ?

• To rationalize the denominator, we simply multiply the numerator and the denominator with the rationalizing factor of the denominator.
• After that, by using algebraic identities, we simplify further.

Explication of steps :

$\longrightarrow \sf { \dfrac{ \sqrt{2} + \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} } = a - b \sqrt{6} }$

Rationalising the denominator in LHS :

• Rationalizing factor of (a - b) is (a + b). So, rationalizing factor of (3√2 - 2√3) is (3√2 + 2√3).

$\longrightarrow \sf { \dfrac{ \sqrt{2} + \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} } \times \dfrac{3 \sqrt{2} + 2 \sqrt{3} }{3 \sqrt{2} + 2 \sqrt{3} } }$

$\longrightarrow \sf { \dfrac{( \sqrt{2} + \sqrt{3} )(3 \sqrt{2} + 2 \sqrt{3})}{(3 \sqrt{2} - 2 \sqrt{3})(3 \sqrt{2} + 2 \sqrt{3}) } }$

As we know that,

• (a - b)(a + b) = a² - b²

$\longrightarrow \sf { \dfrac{( \sqrt{2} )(3 \sqrt{2} + 2 \sqrt{3})+ \sqrt{3}(3 \sqrt{2} + 2 \sqrt{3})}{ {(3 \sqrt{ 2}) }^{2} - { ( 2 \sqrt{3}) }^{2} } }$

$\longrightarrow \sf { \dfrac{6 + 2 \sqrt{ 6} + 3 \sqrt{6} + 6 }{18 - 12} }$

$\longrightarrow \sf { \dfrac{12 + (2 + 3)\sqrt{ 6} }{6} }$

$\longrightarrow \sf { 2 + \dfrac{5 \sqrt{6} }{6} }$

$\longrightarrow \underline{\boxed{\sf { 2 + \dfrac{5}{6} \sqrt{6} } }}$

Now, in RHS we have :

$\longrightarrow \sf { a - b \sqrt{6} }$

From, LHS and RHS, we have :

$\longrightarrow \sf { 2 + \dfrac{5}{6} \sqrt{6} \Leftrightarrow a - b \sqrt{6} }$

$\longrightarrow \underline{\boxed{ { \begin{array}{cc} \sf{a = 2 } \\ \\ \sf b = - \dfrac{5 }{6} \end{array}} }} \: \bigstar$

Answer 2

Step-by-step explanation:

See this Sum carefully Bro