Question

$\frac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } = x + y \sqrt{11}$
Find the value of 'x' and 'y'​

Answer 1

Step-by-step explanation:

$\huge \textbf{ \underline{Answer :-}}$

Given :

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

To Find:

• Find the value of x and y

Solution:

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

Rewrite the LHS and RHS .

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

Now , Rationalize the denominator by 3+2√11

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

Simplify the RHS

By using the formula a²-b²=(a+b)(a-b)

$\implies \tt \bold{ x + y \sqrt{11} = \frac{(5 + \sqrt{11})(3 + 2 \sqrt{11}) }{ {3}^{2} - 4(11)} }$

$\implies \tt \bold{ x + y \sqrt{11} = \frac{15 + 13 \sqrt{11} + 22 }{ - 35} }$

Adding the numbers 15 and 22 is 37 .

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

Now , the comparing the LHS AND RHS.

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

$\therefore \boxed{ \bf{ \red{x = \frac{ - 37}{35}}} }$

$\therefore \boxed{ \red{ \bf{y = \frac{ - 13}{35}}}}$

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Answer 2

Answer:

3−2

11

5+

11

=x+y

11

To Find:

Find the value of x and y

Solution:

\begin{gathered} \implies \tt \bold{ \frac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } = x + y \sqrt{11} } \\ \end{gathered}

⟹

3−2

11

5+

11

=x+y

11

Rewrite the LHS and RHS .

\begin{gathered}\implies \tt \bold{ x + y \sqrt{11} = \frac{5 + \sqrt{11} }{3 - 2 \sqrt{11} } } \\ \end{gathered}

⟹x+y

11

=

3−2

11

5+

11

Now , Rationalize the denominator by 3+2√11

\begin{gathered}\implies \tt \bold{ x + y \sqrt{11} = \frac{5 + \sqrt{11} }{3 - 2 \sqrt{11}} } \bold{ \times \frac{3 + 2 \sqrt{11} }{3 + 2 \sqrt{11} }} \\ \end{gathered}

⟹x+y

11

=

3−2

11

5+

11

×

3+2

11

3+2

11

Simplify the RHS

By using the formula a²-b²=(a+b)(a-b)

\begin{gathered}\implies \tt \bold{ x + y \sqrt{11} = \frac{(5 + \sqrt{11})(3 + 2 \sqrt{11}) }{ {3}^{2} - 4(11)} } \\ \end{gathered}

⟹x+y

11

=

3

2

−4(11)

(5+

11

)(3+2

11

)

\begin{gathered}\implies \tt \bold{ x + y \sqrt{11} = \frac{15 + 13 \sqrt{11} + 22 }{ - 35} } \\ \end{gathered}

⟹x+y

11

=

−35

15+13

11

+22

Adding the numbers 15 and 22 is 37 .

\begin{gathered}\implies \tt \bold{ x + y \sqrt{11} = \frac{37 + 13 \sqrt{11} }{ - 35}} \\ \end{gathered}

⟹x+y

11

=

−35

37+13

11

Now , the comparing the LHS AND RHS.

\begin{gathered}\implies \tt \bold{ x + y \sqrt{11} = \frac{ - 37}{35} - \frac{13}{35} \sqrt{11}} \\ \end{gathered}

⟹x+y

11

=

35

−37

−

35

13

11

\therefore \boxed{ \bf{ \red{x = \frac{ - 37}{35}}} }∴

x=

35

−37

\therefore \boxed{ \red{ \bf{y = \frac{ - 13}{35}}}}∴

y=

35

−13

H

Step-by-step explanation:

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