Question

if A and B are rational numbers and 4 - 3 root 5 upon 4 + 3 root 5 = a + b root 5, find the values of a and b

Answer 1

$\frac{4 - 3 \sqrt{5} } {4 + 3 \sqrt{5} } = a + b \sqrt{5}$

Multiplying by 4-3√5

(4-3√5)²/4²-(3√5)²

= 16+45-24√5/16-45


= 61-24√5/29

= 61/29 - 24/29√5

= a + b√5

So now a = 61/29 , b = -24/29

Answer 2

Answer:

⇒ a = 61/-29

b = 24/-29

Step-by-step explanation:

We have ,

$= > \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} }$

$= > \frac{4 + 3 \sqrt{5} }{(4 - 3 \sqrt{5}) } \times \frac{(4 + 3 \sqrt{5} )}{(4 + 3 \sqrt{5}) }$

$= > \frac{(4 + 3 \sqrt{5})^{2} }{(4)^{2} - (3 \sqrt{5})^{2} }$

$= > \frac{16 + 45 + 2 \times 4 \times 3 \sqrt{5} }{16 - 45}$

$= > \frac{61 + 24 \sqrt{5} }{ - 29} = \frac{61}{ - 29} + \frac{24}{ - 29} \sqrt{5}$

$= > \frac{61 + 24 \sqrt{5} }{ - 29} = \frac{61}{ - 29} + \frac{24}{ - 29} \sqrt{5}$

∴ 4 + 3√5 / 4 - 3√5 = a + b√5

Comparing on both the sides , we get ;

$= > \frac{61}{ - 29} + \frac{24}{ - 29} \sqrt{5} = a + b \sqrt{5}$

$= > a = \frac{61}{ - 29} \: and \: b = \frac{24}{ - 29}$