Question

please do it..............​

Answer 1

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Answer 2

❍ $\Large{\underline{\underline{\sf{Answer:}}}}$

$\huge{\boxed{\sf{\red{\dfrac {10\sqrt{2} - \sqrt{5}}{ 3}}}}}$

❍ $\Large{\underline{\underline{\sf{Step\:by\:step\:explanation:}}}}$

In this question, we need to simplify this equation:

$\displaystyle{\sf{ \frac{3}{2 \sqrt{2} + \sqrt{5}} + \frac{2}{2 \sqrt{2} - \sqrt{5}}}}$

To do so, first we have to

• Rationalize the denomiator of the equation.

but in this case the denominators are conjugate of each other therefore, it is always advisable to find the L.C.M of the denominators.

:$\implies{\sf{ \dfrac{ \big[ 3(2\sqrt{2} - \sqrt{5}) \big] + \big[ 2(2 \sqrt{2} + \sqrt{5}) \big]}{ (2\sqrt{2} + \sqrt{5})(2\sqrt{2} - \sqrt{5})}}}$

• Here a = 2√2, b = √5; a² - b² = (a+b)(a-b), using this identity we will simplify this.

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

:$\implies{\sf{ \dfrac {(6\sqrt{2} - 3\sqrt{5})+ (4 \sqrt{2} + 2\sqrt{5})}{ (2\sqrt{2})^{2} - (\sqrt{5})^{2}}}}$

:$\implies{\sf{ \dfrac {6\sqrt{2} - 3\sqrt{5}+ 4 \sqrt{2} + 2\sqrt{5}}{ (2\sqrt{2})^{2} - (\sqrt{5})^{2}}}}$

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

:$\implies{\sf{ \dfrac {10\sqrt{2} - \sqrt{5}}{ 8- 5}}}$

:$\implies{\boxed{\sf{\red{ \dfrac {10\sqrt{2} - \sqrt{5}}{ 3}}}}}$

And we are done! :D

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➤ What is Rationalization?

➛ The process of converting an irrational number to a rational number.

Denominator of a fraction can be rationalized by multiplying and dividing by its conjugate.

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