Question

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5 } ?} = a + b \sqrt{5}$
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Answer 1

$\huge\mathbb{A~N~S~W~E~R}$

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✔ For this type of question we have to

do this ,

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

$\frac{4 + 3 \ \sqrt{5} }{4 - 3 \sqrt{5} } = a + b \sqrt{5} \\ \\ \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } \times \frac{4 - 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a + b \sqrt{5} \\ \\ \frac{16 - 45}{16 - 24 \sqrt{5} - 45} = a + b \sqrt{5} \\ \\ \frac{ - 29}{ - 29 - 24 \sqrt{5} } = a + b \sqrt{5} \\ \\ - 29 - 29 + 24 \sqrt{5} = a + b \sqrt{5} \\ \\ - 58 + 24 \sqrt{5} = a + b \sqrt{5} \\ \\ \\ a = - 58 \\ \\ b = 24$

${\large\color{Red}{Answer~is~(-58)~and~24 }}$

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Answer 2

$\underline{\underline{\mathbf{\huge{Question}}}}$

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

$\underline{\underline{\mathbf{\huge{Answer}}}}$

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

Rationalising Factor of a-b is a+b

then

$\frac{4 + 3\sqrt{5} \times4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} \times 4 + 3 \sqrt{5} } \\ \\ \frac{(4 + 3 \sqrt{5}) {}^{2} }{(4 {}^{2} ) - (3 \sqrt{5 }) {}^{2} }$

(a+b)²=a²+b²+2ab

(4+3√5)²=4²+(3√5)²+2×4×3√5

=16+9×5+24√5

=16+45+24√5

=61+24√5

a²-b²=(a+b)(a-b)

=4²-3√5²

=16-45

=-29

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

Comparing Sides

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