Question

if a and b are rational numbers and 4+3√5/4-3√5=a+b√5. find the value of a and b.​

Answer 1

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Answer 2

$\huge\text{\underline{Answer}}$

$\sf{\underline{Given }}$

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

we have to find here the value of a and b.

Rationalising the denominator,

$\implies$ $\bold{ \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } \times \frac{4 + 3 \sqrt{5} }{4 + 3 \sqrt{5} } }$

$\implies$ $\bold{\frac{ {(4 +3 \sqrt{5}) }^{2} }{( {4}^{2}) - ( {3 \sqrt{5}) }^{2} } }$

$\implies$ $\bold{ \frac{16 + 24 \sqrt{5} + 45}{16 - 45} }$

$\implies$ $\bold{ \frac{61 + 24 \sqrt{5} }{ - 29} }$

$\implies$ $\bold{ \frac{ - 61 - 24 \sqrt{5} }{29} }$

$\implies$ $\bold{\frac{ - 61}{29} - \frac{24 \sqrt{5} }{29} }$

On comparing with a + b√5 we get

$\implies$ $\bold{a = \frac{ - 61}{29} b = \frac{ - 24 \sqrt{5} }{29} }$