Question

Simplify -$\bf \: \dfrac{3}{ \sqrt{5} \: + \sqrt{2}} \: - \: \dfrac{2}{ \sqrt{7} \: + \sqrt{5}} \: + \: \dfrac{5}{ \sqrt{7} \: - \: \sqrt{2} }$​

Answer 1

Question:-

Question:-To simplify:-

$\sf{\dfrac{3}{\sqrt{5} + \sqrt{2}} - \dfrac{2}{\sqrt{7} + \sqrt{5}} + \dfrac{5}{\sqrt{7} - \sqrt{2}}} </p><p>5</p><p> </p><p> + </p><p>2</p><p> </p><p> </p><p>3</p><p> </p><p> − </p><p>7</p><p> </p><p> + </p><p>5</p><p> </p><p> </p><p>2</p><p> </p><p> + </p><p>7</p><p> </p><p> − </p><p>2</p><p> </p><p> </p><p>5</p><p>$

Solution:-

$\sf{\dfrac{3}{\sqrt{5} + \sqrt{2}} - \dfrac{2}{\sqrt{7} + \sqrt{5}} + \dfrac{5}{\sqrt{7} - \sqrt{2}}\longrightarrow{(i)}} </p><p>5</p><p> </p><p> + </p><p>2</p><p> </p><p> </p><p>3</p><p> </p><p> − </p><p>7</p><p> </p><p> + </p><p>5</p><p> </p><p> </p><p>2</p><p> </p><p> + </p><p>7</p><p> </p><p> − </p><p>2</p><p> </p><p> </p><p>5</p><p> </p><p> ⟶(i)</p><p>$

Let us rationalize all the terms separately,

$\sf{For,\: \dfrac{3}{\sqrt{5} + \sqrt{2}}}For, </p><p>5</p><p> </p><p> + </p><p>2</p><p> </p><p> </p><p>3</p><p>$

By rationalizing the denominator,

$\sf{\dfrac{3}{\sqrt{5} + \sqrt{2}} \times \dfrac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}}= </p><p>5</p><p> </p><p> + </p><p>2</p><p> </p><p> </p><p>3</p><p> </p><p> × </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> </p><p>5</p><p> </p><p> − </p><p>2</p><p>$

$\sf{\dfrac{3(\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})}}= </p><p>( </p><p>5</p><p> </p><p> + </p><p>2</p><p> </p><p> )( </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> )</p><p>3( </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> )$

Applying the identity in the denominator:-

$\boxed{\underline{\red{\rm{(a + b)(a - b) = a^2 - b^2}}}} </p><p> \\ (a+b)(a−b)=a </p><p>2</p><p> −b </p><p>2</p><p>$

$\sf{\dfrac{3(\sqrt{5} - \sqrt{2})}{(\sqrt{5})^2 - (\sqrt{2})^2}}= </p><p>( </p><p>5</p><p> </p><p> ) </p><p>2</p><p> −( </p><p>2</p><p> </p><p> ) </p><p>2</p><p> </p><p>3( </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> )</p><p>$

$=\sf{\dfrac{3(\sqrt{5} - \sqrt{2})}{5 - 2}}= </p><p>5−2</p><p>3( </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> )</p><p>$

$=\sf{\dfrac{3(\sqrt{5} - \sqrt{2})}{3}}= </p><p>3</p><p>3( </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> )$

$=\sf{\dfrac{\not{3}(\sqrt{5} - \sqrt{2})}{\not{3}}}= </p><p></p><p>3</p><p></p><p>3( </p><p>5</p><p> </p><p> − </p><p>2</p><p> </p><p> )$

$\blue{\boxed{\underline{\pink{\rm{\therefore\:\dfrac{3}{\sqrt{5} + \sqrt{2}} = \sqrt{5} - \sqrt{2}}}}}} </p><p> \\ ∴ </p><p>5</p><p> </p><p> + </p><p>2</p><p> </p><p> </p><p>3</p><p> </p><p> = </p><p>5</p><p> </p><p> − </p><p>2$

$\sf{For,\:\dfrac{2}{\sqrt{7} + \sqrt{5}}}For, </p><p>7</p><p> </p><p> + </p><p>5</p><p> </p><p> </p><p>2</p><p>$

By rationalizing the denominator,

$\sf{\dfrac{2}{\sqrt{7} + \sqrt{5}}\times \dfrac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}}}= </p><p>7</p><p> </p><p> + </p><p>5</p><p> </p><p> </p><p>2</p><p> </p><p> × </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> </p><p>7</p><p> </p><p> − </p><p>5</p><p>$

$\sf{\dfrac{2(\sqrt{7} - \sqrt{5})}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})}}= </p><p>( </p><p>7</p><p> </p><p> + </p><p>5</p><p> </p><p> )( </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> )</p><p>2( </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> )$

$\sf{\dfrac{2(\sqrt{7} - \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2}}= </p><p>( </p><p>7</p><p> </p><p> ) </p><p>2</p><p> −( </p><p>5</p><p> </p><p> ) </p><p>2</p><p> </p><p>2( </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> )$

$=\sf{\dfrac{2(\sqrt{7} - \sqrt{5})}{7 - 5}}= </p><p>7−5</p><p>2( </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> )</p><p>$

$=\sf{\dfrac{2(\sqrt{7} - \sqrt{5})}{2}}= </p><p>2</p><p>2( </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> )</p><p>$

$=\sf{\dfrac{\not{2}(\sqrt{7} - \sqrt{5})}{\not{2}}}</p><p> \\ = 22( 7 − 5 )$

$\red{\boxed{\underline{\orange{\rm{\therefore\:\dfrac{2}{\sqrt{7} + \sqrt{5}} = \sqrt{7} - \sqrt{5}}}}}} \\ \\ ∴ </p><p>7</p><p> </p><p> + </p><p>5</p><p> </p><p> </p><p>2</p><p> </p><p> = </p><p>7</p><p> </p><p> − </p><p>5</p><p> </p><p> </p><p>$

$\sf{For,\:\dfrac{5}{\sqrt{7} - \sqrt{2}}}For, </p><p>7</p><p> </p><p> − </p><p>2</p><p> </p><p> </p><p>5</p><p>$

By rationalizing the denominator,

$\sf{\dfrac{5}{\sqrt{7} - \sqrt{2}}\times \dfrac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{7}}}= </p><p>7</p><p> </p><p> − </p><p>2</p><p> </p><p> </p><p>5</p><p> </p><p> × </p><p>7</p><p> </p><p> + </p><p>7</p><p> </p><p> </p><p>7</p><p> </p><p> + </p><p>2$

$\sf{\dfrac{5(\sqrt{7} + \sqrt{2})}{(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})}}= </p><p>( 7 − 2 )( 7 + 2 )5( 7 + 2 )$

$\sf{\dfrac{5(\sqrt{7} + \sqrt{2})}{(\sqrt{7})^2 - (\sqrt{2})^2}}= (7 ) 2 −( 2 ) 25( 7 + 2 )$

$=\sf{\dfrac{5(\sqrt{7} + \sqrt{2})}{7 - 2}}= </p><p>7−25( 7 + 2 )$

$=\sf{\dfrac{\not{5}(\sqrt{7} + \sqrt{2})}{\not{5}}} \\ = 5</p><p>5( </p><p>7</p><p> </p><p> + </p><p>2</p><p> </p><p> )</p><p>$

•55( 7+ 2 )

$\purple{\boxed{\underline{\green{\rm{\therefore\:\dfrac{5}{\sqrt{7} - \sqrt{2}} = \sqrt{7} + \sqrt{2}}}}}}$

∴ 7 − 25 = 7 + 2

Putting all the values in (i)

= (√5 - √2) - (√7 - √5) + (√7 + √2)

= √5 - √2 - √7 + √5 + √7 + √2

= √5 + √5 - √2 + √2 - √7 + √7

= 2√5

$\boxed{\underline{\rm{\red{\therefore\:\dfrac{3}{\sqrt{5} + \sqrt{2}} - \dfrac{2}{\sqrt{7} + \sqrt{5}} + \dfrac{5}{\sqrt{7} - \sqrt{2}} = 2\sqrt{5}}}}}$

∴ 5 +23 − 7 + 52 + 7 − 25 =25

______________________________________

Answer 2

Given Expression:

$\sf{\dfrac{3}{\sqrt{5}+\sqrt{2}}-\dfrac{2}{\sqrt{7}+\sqrt{5}}+\dfrac{5}{\sqrt{7}-\sqrt{2}}$

Solution:

In order to simplify the expression, we need to rationalize each term:

$\sf{=\dfrac{3}{\sqrt{5}+\sqrt{2}}\times\dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}-\dfrac{2}{\sqrt{7}+\sqrt{5}}\times\dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\dfrac{5}{\sqrt{7}-\sqrt{2}}\times\dfrac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}}$

$\sf{=\dfrac{3\sqrt{5}-3\sqrt{2}}{5-2}-\dfrac{2\sqrt{7}-2\sqrt{5}}{7-5}+\dfrac{5\sqrt{7}+5\sqrt{2}}{7-2}}$

$\sf{=\dfrac{3(\sqrt{5}-\sqrt{2})}{3}-\dfrac{2(\sqrt{7}-\sqrt{5})}{2}+\dfrac{(\sqrt{7}+\sqrt{2})}{5}}$

Cancelling the like terms, we are left with:

$\sf{=\sqrt{5}-\sqrt{2}-\sqrt{7}+\sqrt{5}+\sqrt{7}+\sqrt{2}}$

Cancelling the like terms with negative signs:

$\sf{=\sqrt{5}+\sqrt{5}}$

$\bf{=2\sqrt{5}}$

Hence the simplified answer is 2√5.

Rationalizing the denominator:

To rationalize the denominator of a fraction, we are removing the rational numbers from the denominator by multiply it with the terms of denominator with the opposite sign.