Question

Rationalize the denominator of:-

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

Answer 1

Answer:

Question :-

Rationalize the denominator of :-

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

Answer :-

• Let us solve this question in two steps.

Step 1 :- Multiply and divide the expression by (4 - √2 + √5)

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

$\sf : \implies {\dfrac{4\ -\ \sqrt{2}\ -\ \sqrt{5}}{16\ -\ (2\ +\ 5\ +\ 3\ \sqrt{10})}}$

$\sf : \implies {\dfrac{4\ -\ \sqrt{2}\ -\ \sqrt{5}}{9\ -\ 2 \sqrt{10}}}$

Step 2 :- Multiply and divide the expression by the conjugate of the denominator. i.e., by (9 + 2√10)

$\sf : \implies {\dfrac{4\ -\ \sqrt{2}\ -\ \sqrt{5}}{9\ -\ 2 \sqrt{10}} \times \dfrac{9\ +\ 2 \sqrt{10}}{9\ +\ 2 \sqrt{10}}}$

$\sf : \implies {\dfrac{36\ +\ 8 \sqrt{10}\ -\ 9 \sqrt{2}\ -\ 2 \sqrt{20}\ -\ 9 \sqrt{5}\ -\ 2 \sqrt{50}}{81\ -\ 40}}$

$\sf : \implies {\dfrac{36\ +\ 8 \sqrt{10}\ -\ 9 \sqrt{2}\ -\ 4 \sqrt{5}\ -\ 9 \sqrt{5}\ -\ 10 \sqrt{2}}{41}}$

$\sf : \implies {\dfrac{36\ +\ 8 \sqrt{10}\ -\ 19 \sqrt{2}\ -\ 13 \sqrt{5}}{41}}$

${\therefore{\underline{\boxed{\sf {\dfrac{1}{4\ +\ \sqrt{2}\ +\ \sqrt{5}}\ =\ \dfrac{36\ +\ 8 \sqrt{10}\ -\ 19 \sqrt{2}\ -\ 13 \sqrt{5}}{41}}}}}}$

Answer 2

Step-by-step explanation:

Rationalize the denominator of:-

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