Question

If a=8+3√7 and b=1/a, what will be the value of a²+b²?

Answer 1

$\huge\blue{Heya}$

$\textbf{Given :-}$

a = 8 + 3√7

b = 1/a

b = 1/(8 + 3√7)

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

Now,.

$\textbf{To find :-}$

a² + b²

= (8 + 3√7)² + (8 - 3√7)²

= (8)² + (3√7)² + 2(8)(3√7) + (8)² + (3√7)² - 2(8)(3√7)

= 64 + 63 + 48√7 + 64 + 63 - 48√7

= 254

$\textbf{Hope this helps you.}$

Answer 2

given :-

a = 8 + 3√7

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


ATQ, b = 1/a

hence, the value of b is = 8 - 3√7

now we have to find a² + b²

= (8 +3√7)² + (8 - 3√7)²

= (8)² + 2(8)(3√7) + (3√7)² + (8)² - 2(8)(3√7) + (3√7)²

= 64 + 48√3 + 63 + 64 - 48√7 + 63

= 254

HOPE THIS HELPS..!!