Question

if (root5+root3)/(root5-root3)=a+root15b,find the value of a and b

Answer 1

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5}- \sqrt{3} } = a+b \sqrt{15}$

By Rationalization,

L.H.S,

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} }$×$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5}+ \sqrt{3} }$

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On Nominator,

by formula, $(a+b)^2 =a^2+ b^2+2ab$

On Denominator,

by formula, $a^2 -b^2 =(a+b)(a-b)$
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$\frac{ (\sqrt{5}+ \sqrt{3})^2 }{ (\sqrt{5})^2 - ( \sqrt{3})^2 }$

$\frac{5 + 3 + 2 \sqrt{15} }{5-3}$

$\frac{8+ 2\sqrt{15} }{2}$

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Then,

$\frac{8 + 2 \sqrt{15} }{2} = a+b \sqrt{15}$

$\frac{8}{2} + \frac{2 \sqrt{15} }{2} = a+b \sqrt{15}$

$4 + \sqrt{15} = a+b \sqrt{15}$

Now,

a =4
b√15 =√15

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a= 4 

b =1 


i hope this will help you


-by ABHAY

Answer 2

Hey there

√5 + √3 / √5 - √3 = a + √15b

Rationalising the denominator ; Multiplying denominator and numerator by √5 + √3

(√5 + √3)² / (√5)²-(√3)² = a + √15b

5 + 3 + 2√15 / 5- 3 = a + √15b

8+2√15 / 2 = a + √15b

4 + √15 = a + √15 b.

So,

Comparing equations,

We get a = 4 , b = 1 .

Hope helped!