Question

If √9+√77,then find the value of 1/11(x+2/x)²

Answer 1

Answer:

53

hope it works

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Answer 2

If x=√(9+√77) then the value of [(x+2/x)^2]/11 is?​

The value of [(x+2/x)^2]/11  is 2.

Given:

x=√(9+√77)

To Find:

The value of [(x+2/x)^2]/11

Solution:

$x = \sqrt{(9+\sqrt{77} }$

Squaring  on both sides

$x^{2} = (\sqrt{(9+\sqrt{77} } )^{2} \\\\x^{2} = 9 +\sqrt{77}------------(1)$

reciprocal on both side

$\frac{1}{x^{2} } = \frac{1}{9+\sqrt{77} }$

Rationalizing R.H.S

$\frac{1}{x^{2} } = \frac{1}{9+\sqrt{77} } * \frac{9-\sqrt{77}}{9-\sqrt{77}} \\\\\frac{1}{x^{2} } = \frac{9-\sqrt{77}}{9^{2} -\sqrt{77} ^{2} } \\\\\frac{1}{x^{2} } = \frac{9-\sqrt{77}}{81-77} \\\\\frac{1}{x^{2} } = \frac{9-\sqrt{77}}{4}-------------(2)$

Consider (x+ 2/x)²

This is of the form (a+b)²= a² +2ab +b²,where a = x and b = 2/x .

$=x^{2} + 2*x * \frac{2}{x} +( \frac{2}{x})^{2} \\\\=x^{2} +4+4*\frac{1}{x^{2} }$

Substituting the value of x² and 1/x² from equation (1)

$=9+\sqrt{77} +4+4*\frac{9-\sqrt{77} }{4 }\\\\=9+\sqrt{77} +4+9-\sqrt{77}\\\\= 22$

Now ,

[(x+2/x)^2]/11

= 22/11

=2

∴ The value of [(x+2/x)^2]/11  is 2.

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