Question

if a=8+3√7and b=1/a than what will be the value of a square + b square
$a {}^{2} + b {}^{2}$
​

Answer 1

Answer:

254.

Step-by-step explanation:

Given,

a = 8 + 3√7

So,

$b = \frac{1}{a} \\ \\ b = \frac{1}{8 + 3 \sqrt{7} }$

Rationalizing the denominator we get:

$b = \frac{1}{8 + 3 \sqrt{7} } \times \frac{8 - 3 \sqrt{7} }{8 - 3 \sqrt{7} } \\ \\ b = \frac{8 - 3 \sqrt{7} }{ {(8)}^{2} - {(3 \sqrt{7}) }^{2} } \\ \\ b = \frac{8 - 3 \sqrt{7} }{64 - 63} \\ \\ \bf b = 8 - 3 \sqrt{7}$

Putting the value in a² + b²

${(8 + 3 \sqrt{7}) }^{2} + {(8 - 3 \sqrt{7} )}^{2} \\ \\ = ( {(8)}^{2} + {(3 \sqrt{7}) }^{2} + 2.8.3 \sqrt{7} ) \\ + ( {(8)}^{2} + {(3 \sqrt{7} }^{2} ) - 2.8.3 \sqrt{7} ) \\ \\ = (64 + 63 + 48 \sqrt{7} ) + (64 + 63 - 48 \sqrt{7} ) \\ \\ = 127 + 48 \sqrt{7} + 127 - 48 \sqrt{7} \\ \\ \bf = 254$