Question

root 8 + root 7 upon root 8 minus root 7 Y is equal to root 8 minus root 7 upon root 8 + root 7 find the value of x square + y square minus 3xy

Answer 1

Hey friend,
Here is the answer you were looking for:
$x = \frac{ \sqrt{8} + \sqrt{7} }{ \sqrt{8} - \sqrt{7} }$
On rationalizing the denominator we get:

$x = \frac{ \sqrt{8} + \sqrt{7} }{ \sqrt{8} - \sqrt{7} } \times \frac{ \sqrt{8} + \sqrt{7} }{ \sqrt{8} + \sqrt{7} }$

Using the identities:

$(a + b)^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

$x = \frac{ {( \sqrt{8} })^{2} + {( \sqrt{7} )}^{2} + 2( \sqrt{8} )( \sqrt{7} ) }{ {( \sqrt{8}) }^{2} - {( \sqrt{7} })^{2} } \\ \\ x = \frac{8 + 7 + 2 \sqrt{56} }{8 - 7} \\ \\ splitting \: 56 \\ \\ x = 15 + 2 \sqrt{2 \times 2 \times 2 \times 7} \\ \\ x = 15 + 2 \times 2 \sqrt{14} \\ \\ x = 15 + 4 \sqrt{14}$

$y = \frac{ \sqrt{8} - \sqrt{7} }{ \sqrt{8} + \sqrt{7} }$

On rationalizing the denominator we get:

$y = \frac{ \sqrt{8} - \sqrt{7} }{ \sqrt{8} + \sqrt{7} } \times \frac{ \sqrt{8} - \sqrt{7} }{ \sqrt{8} - \sqrt{7} }$

Using the identities:

${(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

$y = \frac{ {( \sqrt{8} )}^{2} + {( \sqrt{7} )}^{2} - 2( \sqrt{8})( \sqrt{7} ) }{ {( \sqrt{8} })^{2} - {( \sqrt{7} )}^{2} } \\ \\ y = \frac{8 + 7 - 2 \sqrt{56} }{8 - 7} \\ \\ (splitting \: 56) \\ \\ y = 15 - 2 \sqrt{2 \times 2 \times 2 \times 7} \\ \\ y = 15 - 2 \times 2 \sqrt{14} \\ \\ y = 15 - 4 \sqrt{14}$
${x}^{2} + {y}^{2} - 3xy$


Putting the values:

${(15 + 4 \sqrt{14} )}^{2} + {(15 - 4 \sqrt{14}) }^{2} - 3(15 + 4 \sqrt{14} )(15 - 4 \sqrt{14} )$

Using same identities:

$( {(15)}^{2} + {(4 \sqrt{14}) }^{2} + 2(15)(4 \sqrt{14} )) + ( {(15)}^{2} + {(4 \sqrt{14}) }^{2} - 2(15)(4 \sqrt{14} )) - 3( {(15)}^{2} - {(4 \sqrt{14} )}^{2} ) \\ \\ =( 225 + 224 + 120 \sqrt{14} ) + (225 + 224 - 120 \sqrt{14} ) - 3(225 - 224) \\ \\ = (449 + 120 \sqrt{14} ) + (449 - 120 \sqrt{14} ) - 3(1) \\ \\ = 449 + 120 \sqrt{14} + 449 - 120 \sqrt{14} - 3 \\ \\ ( + 120 \sqrt{14} \: and \: - 120 \sqrt{14} \: got \: cancel) \\ \\ = 898 - 3 \\ \\ = 895$

Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
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