Question

Please Tell me the answers fast urgent ​

Answer 1

SOLUTION.

Here,

$\tt \implies x = \dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

$\tt \implies y = \dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}$

Rationalizing x, we get,

$\tt \implies x = \dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}} \times \dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}$

$\tt \implies x = \dfrac{(\sqrt{5}+\sqrt{2})^{2}}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})}$

$\tt \implies x = \dfrac{5+2+2\sqrt{10}}{5-2}$

$\tt \implies x = \dfrac{7+2\sqrt{10}}{3}$

Rationalizing y, we get,

$\tt \implies y = \dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}} \times \dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

$\tt \implies y = \dfrac{(\sqrt{5}-\sqrt{2})^{2}}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}$

$\tt \implies y = \dfrac{5+2-2\sqrt{10}}{5-2}$

$\tt \implies y = \dfrac{7-2\sqrt{10}}{3}$

So,

$\tt 3x^{2} +4xy - 3y^{3}$

$\tt= 3x^{2} +4x \times \dfrac{1}{x}- 3y^{3}$

$\tt= 3x^{2} + 4- 3y^{3}$

$\tt = 3 \times \bigg(\dfrac{7+2\sqrt{10}}{3}\bigg)^{2} - 3\times \bigg(\dfrac{7-2\sqrt{10}}{3}\bigg)^{3} +4$

$\tt = \dfrac{3(7+2\sqrt{10})^{2}}{9} - \dfrac{(7-2\sqrt{10})^{3}}{9} + 4$

$\tt = \dfrac{3(49+40+28\sqrt{10})}{9} - \dfrac{(7)^{3}-(2\sqrt{10})^{3}-3\times7^{2}\times2\sqrt{10}+3\times7\times(2\sqrt{10})^{2}}{9}+4$

$\tt = \dfrac{3(89+28\sqrt{10})}{9} - \dfrac{1183-374\sqrt{10}}{9} +4$

$\tt = \dfrac{267+84\sqrt{10}}{9} - \dfrac{1183-374\sqrt{10}}{9}+4$

$\tt = \dfrac{267+84\sqrt{10}-1183+374\sqrt{10}}{9} + 4$

$\tt = \dfrac{458\sqrt{10}-916}{9} +4$

$\tt = \dfrac{458\sqrt{10}-916+36}{9}$

$\tt = \dfrac{458\sqrt{10}-880}{9}$

Which is our required answer.

Answer 2

Question :

$\sf{If \: x \small \implies \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } } \: and \: y \small \: \implies \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \: find \: 3 {x}^{3} + 4xy - {3y}^{3}$

Solution :

First we will rationalize the denominators.

So we have rationalize the denominator in variable x.

$\rightarrow \sf \: x = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} }$

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

$\sf \: \rightarrow \: x = \frac{ {( \sqrt{5} + \sqrt{2} ) }^{2} }{ {( \sqrt{5}) }^{2} - {( \sqrt{2}) }^{2} }$

$\sf \: \rightarrow \: x = \frac{ {( \sqrt{5}) }^{2} + 2 \cdot \sqrt{5} \cdot \sqrt{2} + {( \sqrt{2}) }^{2} }{5 - 2}$

$\sf \: \rightarrow \: x = \frac{5 + 2 \sqrt{10} + 2 }{3}$

$\sf \: \rightarrow \: x = \frac{7 + 2 \sqrt{10} }{3}$

Now, we will rationalize the denominator in variable y.

$\sf \: \rightarrow \:y = \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} }$

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

$\sf \: \rightarrow \: y = \frac{ {( \sqrt{5} - \sqrt{2} ) }^{2} }{ {( \sqrt{5}) }^{2} - {( \sqrt{2}) }^{2} }$

$\sf \: \rightarrow \: x = \frac{ {( \sqrt{5}) }^{2} - 2 \cdot \sqrt{5} \cdot \sqrt{2} + {( \sqrt{2}) }^{2} }{5 - 2}$

$\sf \: \rightarrow \: x = \frac{5 - 2 \sqrt{10} + 2 }{3}$

$\sf \rightarrow y = \frac{7 - 2 \sqrt{10} }{3}$

Now, we have rationalized the denominators and we will put the values in place of variables :-

3x³– 4xy + 3y³

$\sf \: \rightarrow \: 3 { (\frac{7 + 2 \sqrt{10} }{3}) }^{3} - 4 (\frac{7 + 2 \sqrt{10} }{3})( \frac{7 - 2 \sqrt{10} }{3} ) + 3 { (\frac{7 - 2 \sqrt{10} }{3} )}^{3}$

$\small \: \sf \: \rightarrow \: \frac{3 \times ({7})^{3} + 3 \cdot7 \cdot2 \sqrt{10}(7 + 2 \sqrt{10} ) + 3 \times ({2 \sqrt{10}})^{3} + 3 \times ( {7})^{3} - 3 \cdot7 \cdot2 \sqrt{10}(7 - 2 \sqrt{10} ) - 3 \times ({2 \sqrt{10}})^{3} }{27} + 4$

$\small \sf \: \rightarrow \frac{3 \times 343 + 42 \sqrt{10}(7 + 2 \sqrt{10} ) + 240 \sqrt{10} + 3 \times 343 - 42 \sqrt{10} (7 - 2 \sqrt{10} ) - 240 \sqrt{10} }{27} + 4$

$\small \sf \rightarrow \: \frac{1029 + 294 \sqrt{10} + 840 + 1029 - 294 \sqrt{10} + 840}{27} + 4$

$\frac{\small \sf \rightarrow 1029 + 1029 + 840 + 840}{27} + 4$

$\sf \: \rightarrow \frac{2058 + 1680}{27} + 4$

$\frac{3738}{27} + \frac{108}{27}$

$\frac{3738 + 108}{27}$

$\frac{3846}{27}$

$142.44$

Hence the answer is 142.44