Question

find a and b such that 2+5√7/2-5√7=a+b√7

Answer 1

Answer:

= [(2 + 5√7)/(2 – 5√7)] × [(2 + 5√7)/(2 + 5√7)]

= [ (4 + 175 + 20√7)/(4 – 175)]

= [(179 + 20√7)/(-171)]

= a + b√7

So, a = - 179/171

b = - 20/171

Answer 2

Step-by-step explanation:

Given:

$\dfrac{2 + 5 \sqrt{7} }{2 - 5 \sqrt{7} } = a + b\sqrt{7}$

To find:

The value of a and b

Answer in 3 steps

1. Rationalize the denominator.

2. Use Algebra Formula

3. Simplify and get answer.

Solution:

L.H.S

$\dfrac{2 + 5 \sqrt{7} }{2 - 5 \sqrt{7} } \\ \\ = \frac{(2 + 5 \sqrt{7})(2 + 5 \sqrt{7})}{(2 - 5 \sqrt{7})(2 + 5 \sqrt{7})} \\ \\ = \frac{ {(2)}^{2} + {(5 \sqrt{7})}^{2} + 2.2.5 \sqrt{7} }{ {(2)}^{2} - {(5 \sqrt{7} }^{2})} \\ \\ = \frac{4 + 25.7+ 20\sqrt{7} }{4 - 25.7} \\ \\ = \frac{4 + 175 + 20 \sqrt{7} }{ 4 - 175} \\ \\ = \frac{179 + 20 \sqrt{7} }{ - 171} \\ \\ = - \frac{179}{171} - \frac{20\sqrt{7} }{171}$

Comparing L.H.S from R.H.S I get,

$a = - \dfrac{179}{171}$

$b\sqrt{7} = - \dfrac{20\sqrt{7} }{171} \implies b = - \dfrac{20 }{171}$

Used Algebra Formula

${(x + y)}^{2}={x}^{2}+{y}^{2}+2xy\\ \\{(x - y)}^{2}={x}^{2}+{y}^{2}-2xy\\ \\(x+y)(x-y) = x^2 - y^2$