Question

find the value of x and y..
show the full calculation ​

Answer 1

Given :-

$\sf{\implies \frac{5 + \sqrt{11} }{3 - 2\sqrt{11} } = x + y\sqrt{11} }$

To Find :-

• Value of x and y .

Solution :-

Taking LHS firstly :-

$\sf{\implies \; LHS = \frac{5 +\sqrt{11} }{3 - 2\sqrt{11} } }$

Now rationalising

$\sf{\implies \frac{5 +\sqrt{11} }{3 - 2\sqrt{11}} \times \frac{3 +2\sqrt{11} }{3 + 2\sqrt{11}} }$

$\sf{\implies \; Numerator = 5+\sqrt{11} \times (3+2\sqrt{11})= 15 + 3\sqrt{11} + 10\sqrt{11} + 22 }$

$\sf{\implies \; Numerator = 15 + \sqrt{11}(3 + 10) + 22 = 15 + 13\sqrt{11} + 22 }$

$\sf{\implies \; Denominator = 3 -2\sqrt{11} \times (3 + 2\sqrt{11} )}$

$\sf{\implies \; Denominator = (3)^{2} - (2\sqrt{11} )^{2} }$

$\sf{\implies \; Denominator = 9 - 44 = -35}$

$\sf{\implies \; LHS = \frac{37 + 13\sqrt{11} }{-35} }$

Comparing both sides we got

$\sf{\implies \frac{37 + 13\sqrt{11} }{-35 } = x + y\sqrt{11} }$

$\sf{\implies x =( \frac{-37}{35}) \; and y = (- \frac{13}{35} ) }$

Answer 2

Question :–

• If x and y are rational numbers and $\: { \bold{ \dfrac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } = x + y \sqrt{11} }} \:$ , Find the values of x and y.

$\\ \rule{220}{2}$

ANSWER :–

GIVEN :–

$\\ \: { \huge{.}} \: \: \: \: { \bold{ \dfrac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } = x + y \sqrt{11} }} \:$

TO FIND :–

▪︎ Value of x and y .

SOLUTION :–

$\\ \implies{ \bold{ \dfrac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } = x + y \sqrt{11} }} \:$

• Now rationalization –

$\\ \implies{ \bold{ \dfrac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } \times \dfrac{3 + 2 \sqrt{11} }{3 + 2 \sqrt{11} } = x + y \sqrt{11} }} \:$

$\\ \implies{ \bold{ \dfrac{(5 + \sqrt{11} )(3 + 2 \sqrt{11} )}{(3 - 2 \sqrt{11})(3 + 2 \sqrt{11} )} = x + y \sqrt{11} }} \:$

• using identity –

$\\ \implies \large { \pink{ \boxed{ \bold{ (a + b)(a - b) = {a}^{2} - {b}^{2} }}}} \:$

• So that –

$\\ \implies{ \bold{ \dfrac{15 + 10 \sqrt{11} + 3 \sqrt{11} + 2(11)}{{3 }^{2} - {(2 \sqrt{11} )}^{2}} = x + y \sqrt{11} }} \:$

$\\ \implies{ \bold{ \dfrac{15 + 13 \sqrt{11} + 2(11)}{9 - 44} = x + y \sqrt{11} }} \:$

$\\ \implies{ \bold{ \dfrac{37+ 13 \sqrt{11} }{9 - 44} = x + y \sqrt{11} }} \:$

$\\ \implies{ \bold{ \dfrac{37+ 13 \sqrt{11} }{ - 35} = x + y \sqrt{11} }} \:$

$\\ \implies{ \bold{ - \frac{37}{35} - \frac{13}{35} \sqrt{11} = x + y \sqrt{11} }} \:$

• Now compare –

$\\ \implies \large{ \red{ \bold{x = - \frac{37}{35} \: \: and\: \: y = - \frac{13}{35} }}} \:$

$\\ \rule{220}{2}$