Question

» Rationalize the denominator
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Answer 1

$\red{\large\mathfrak{Question}}$

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$\sf \frac{1}{4 + \sqrt{2} + \sqrt{5} } \:$

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$\green{\large\mathfrak{answer}}$

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$\sf \: = \frac{1}{4 + ( \sqrt{2} + \sqrt{5} ) }$

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Now rationalizing the denominator

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$\sf \: = \: \frac{1}{4 + ( \sqrt{2} + \sqrt{5} ) } \times \frac{ ( \sqrt{2 } - \sqrt{5}) }{ ( \sqrt{2} - \sqrt{5}) }$

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$\sf \: = \: \frac{1 \times ( \sqrt{2} - \sqrt{5} )}{4 + (( { \sqrt{2}) }^{2} - ( { \sqrt{5} )}^{2} )}$

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$\sf \: = \frac{ \sqrt{2 } - \sqrt{5} }{4 + (2 - 5)}$

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$\sf \: = \: \frac{ \sqrt{2} - \sqrt{5} }{4 + (2 - 5)}$

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$\sf \: \: = \: \frac{ \sqrt{2} - \sqrt{5} }{(4 - 3)}$

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$= \sf \: \frac{ \sqrt{2} - \sqrt{5} }{1}$

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$= \sf \: \sqrt{2 } - \sqrt{5}$

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$\pink{\large\mathfrak{hope \: it \: helps \: you}}$

Answer 2

Answer:

(4 - √2 + √5)(9 + 2√10)/41

Step-by-step explanation:

Rationalize the denominator means making the denominator a rational number. In order to rationalize the denominator, we multiply the given fraction with the rationalizing factor of the denominator with both the numerator and the denominator of the given fraction.

Here, we have to rationalize the denominator of :

$\begin{gathered} \twoheadrightarrow \rm \frac{1}{4 + \sqrt{2} + \sqrt{5} } \: \\ \end{gathered}$

Rationalizing factor of a term means changing the sign to its opposite sign. So, the rationalizing factor of (4 + √2 + √5) is (4 - √2 + √5). Multiplying (4 - √2 + √5) with both the numerator and the denominator.

$\begin{gathered} \twoheadrightarrow \rm \frac{1}{4 + \sqrt{2} + \sqrt{5} } \: \\ \end{gathered} \times \frac{4 - \sqrt{2} + \sqrt{5} }{4 - \sqrt{2} + \sqrt{5} }$

Rearranging the terms.

$\begin{gathered} \twoheadrightarrow \rm \frac{1(4 - \sqrt{2} + \sqrt{5})}{(4 + \sqrt{2} + \sqrt{5} )( 4 - \sqrt{2} + \sqrt{5}) } \: \\ \end{gathered}$

Multiplying 1 with the terms in the brackets in the numerator & Simplifying further in the denominator,

$\begin{gathered} \twoheadrightarrow \rm \frac{4 - \sqrt{2} + \sqrt{5}}{(16 - {(\sqrt{2} + \sqrt{5})}^{2} ) } \\ \end{gathered}$

Putting the value of square of the number in the denominator.

$\begin{gathered} \twoheadrightarrow \rm \frac{4 - \sqrt{2} + \sqrt{5}}{16 - (2 + 5 + 2 \sqrt{10} ) } \\ \end{gathered}$

Adding & subtracting the numbers in the denominator,

$\begin{gathered} \twoheadrightarrow \rm \frac{4 - \sqrt{2} + \sqrt{5}}{9- 2 \sqrt{10} } \\ \end{gathered}$

Here, the rationalizing factor of (9 - 2√10) is (9 + 2√10). Multiplying (9 + 2√10) with both the numerator and the denominator.

$\begin{gathered} \twoheadrightarrow \rm \dfrac{4 - \sqrt{2} + \sqrt{5}}{9- 2 \sqrt{10} } \times \dfrac{9 + 2 \sqrt{10} }{9 + 2 \sqrt{10} } \\ \end{gathered}$

Rearranging the terms.

$\begin{gathered} \twoheadrightarrow \rm \dfrac{(4 - \sqrt{2} + \sqrt{5})( 9 + 2 \sqrt{10} ) }{(9- 2 \sqrt{10})( 9 + 2 \sqrt{10} )} \\ \end{gathered}$

Using the identity given below in the denominator.

• (a + b)(a - b) = a² - b²

$\begin{gathered} \twoheadrightarrow \rm \dfrac{(4 - \sqrt{2} + \sqrt{5})( 9 + 2 \sqrt{10} ) }{{(9)}^{2} -{ (2 \sqrt{10})}^{2} } \\ \end{gathered}$

Putting the value of square of the number in the denominator.

$\begin{gathered} \twoheadrightarrow \rm \dfrac{(4 - \sqrt{2} + \sqrt{5})( 9 + 2 \sqrt{10} ) }{81 -40 } \\ \end{gathered}$

Subtracting the numbers in the denominator.

$\begin{gathered} \twoheadrightarrow \rm \dfrac{(4 - \sqrt{2} + \sqrt{5})( 9 + 2 \sqrt{10} ) }{41 } \\ \end{gathered}$

∴ Hence, Denominator is rationalized !