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Hint: Put the value of $x$ directly in the given expression and find out its value.cylindrical portion of the pillar is conical

Given, $x = 2\sqrt 6 + 5$ and we have to find the value of $x + \dfrac{1}{x}$.

Let $x + \dfrac{1}{x} = y,$then we have:

$ \Rightarrow y = \dfrac{{{x^2} + 1}}{x}$

Now, putting the value of$x$, we’ll get:

$ \Rightarrow y = \dfrac{{{{(2\sqrt 6 + 5)}^2} + 1}}{{(2\sqrt 6 + 5)}}$

We know that${(a + b)^2} = {a^2} + {b^2} + 2ab$, using this formula we’ll get:

$

\Rightarrow y = \dfrac{{\left[ {{{(2\sqrt 6 )}^2} + {5^2} + 2 \times 2\sqrt 6 \times 5} \right] + 1}}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = \dfrac{{24 + 25 + 20\sqrt 6 + 1}}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = \dfrac{{50 + 20\sqrt 6 }}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = \dfrac{{10\left( {2\sqrt 6 + 5} \right)}}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = 10 \\

$

Thus, the required value of $x + \dfrac{1}{x} $ is $10$.

Note: If we are getting an irrational number in a denominator in any expression, then we can convert it into a rational number by rationalizing it. In this process we multiply both numerator and denominator by the conjugate of the denominator.

Given, $x = 2\sqrt 6 + 5$ and we have to find the value of $x + \dfrac{1}{x}$.

Let $x + \dfrac{1}{x} = y,$then we have:

$ \Rightarrow y = \dfrac{{{x^2} + 1}}{x}$

Now, putting the value of$x$, we’ll get:

$ \Rightarrow y = \dfrac{{{{(2\sqrt 6 + 5)}^2} + 1}}{{(2\sqrt 6 + 5)}}$

We know that${(a + b)^2} = {a^2} + {b^2} + 2ab$, using this formula we’ll get:

$

\Rightarrow y = \dfrac{{\left[ {{{(2\sqrt 6 )}^2} + {5^2} + 2 \times 2\sqrt 6 \times 5} \right] + 1}}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = \dfrac{{24 + 25 + 20\sqrt 6 + 1}}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = \dfrac{{50 + 20\sqrt 6 }}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = \dfrac{{10\left( {2\sqrt 6 + 5} \right)}}{{(2\sqrt 6 + 5)}}, \\

\Rightarrow y = 10 \\

$

Thus, the required value of $x + \dfrac{1}{x} $ is $10$.

Note: If we are getting an irrational number in a denominator in any expression, then we can convert it into a rational number by rationalizing it. In this process we multiply both numerator and denominator by the conjugate of the denominator.

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