Question

$1. \: \sf\frac{4 + \sqrt{5} }{4 - \sqrt{5}} + \frac{4 - \sqrt{5} }{4 + \sqrt{5} }$
$\sf \: 2. \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$
$\sf \: 3. \frac{3}{5 - \sqrt{3} } + \frac{2}{5 + \sqrt{3} }$
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Answer 1

Answer:-

$\sf \: 1. \: \: \dfrac{4 + \sqrt{5} }{4 - \sqrt{5} } + \dfrac{4 - \sqrt{5} }{4 + \sqrt{5} } \\ \\ \\ \implies \sf \: \frac{(4 + \sqrt{5}) ^{2} + (4 - \sqrt{5} ) ^{2} }{(4 - \sqrt{5})(4 + \sqrt{5} ) }$

Using (a + b)² + (a - b)² = 2(a² + b²) and (a + b)(a - b) = a² - b² we get,

$\implies \sf \: \frac{2 \big( {4}^{2} + (\sqrt{5}) ^{2} \big)}{ {4}^{2} - {( \sqrt{5} )}^{2} } \\ \\ \\ \implies \sf \: \frac{2(16 + 5)}{16 - 5} \\ \\ \\ \implies \sf \: \frac{2 \times 21}{11} \\ \\ \\ \implies \red{\sf \: \frac{42}{11} }$

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$\sf \: 2. \: \: \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2}}$

Rationalise the denominator.

$\: \implies \sf \: \dfrac{( \sqrt{3} + \sqrt{2})( \sqrt{3} + \sqrt{2}) }{( \sqrt{3} - \sqrt{2})( \sqrt{3} + \sqrt{2}) } \\ \\ \\ \implies \sf \: \frac{{( \sqrt{3} + \sqrt{2}) }^{2} }{ {( \sqrt{3} )}^{2} - ({\sqrt{2}})^{2} }$

Using (a + b)² = a² + b² + 2ab we get,

$\implies \sf \: \frac{ {( \sqrt{3}) }^{2} + ({\sqrt{2}})^{2} + 2 \times \sqrt{3} \times \sqrt{2} }{3 - 2} \\ \\ \\ \implies \sf \: \frac{3 + 2 + 2 \sqrt{2 \times 3} }{1} \:\:\:\: (\because \sqrt{a} \times \sqrt{b} = \sqrt{ab} )\\ \\ \\ \implies \red{\sf \: 5+ 2 \sqrt{6} ) }$

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$\: \implies \sf \: \frac{3}{5 - \sqrt{3} } + \frac{2}{5 + \sqrt{3} } \\ \\ \\ \implies \sf \: \frac{3(5 + \sqrt{3} ) + 2(5 - \sqrt{3} )}{(5 - \sqrt{3} )(5 + \sqrt{3}) } \\ \\ \\ \implies \sf \: \frac{15 + 3 \sqrt{3} + 10 - 2 \sqrt{3} }{ {5}^{2} - {( \sqrt{3} )}^{2} } \\ \\ \\ \implies \sf \: \frac{25 + \sqrt{3} }{25 - 3} \\ \\ \\ \implies \red{\sf \: \frac{25 + \sqrt{3} }{22} }$

Answer 2

Given :-

1. 4 + √5/4 - √5 + 4 - √5/4 + √5

2. √3 + √2/√3 - √2

3. 3/5 - √3 + 2/5 + √3

To Find :-

Value

Solution :-

1]

By taking LCM

(4 + √5)² + (4 - √5)²/(4 - √5)(4 + √5)

2(4² + √5)²/(4)² - (√5)²

2(16 + 5)/16 - 5

2(21)/11

42/11

2]

√3 + √2/√3 - √2 × √3 + √2/√3 + √2

√3 + √2 × √3 + √2/√3 - √2 × √3 + √2

(√3 + √2)²/(√3)² - (√2)²

• (a + b)² = a² + b² + 2ab

(√3)² + (√2)² + 2(√3)(√2)/3 - 2

3 + 2 + 2√6/1

5 + 2√6

3]

3(5 + √3) + 2(5 - √3)/(5 - √3)(5 + √3)

15 + 3√3 + 10 - 2√2/(5)² - (√3)²

25 + √3/25 - 3

25 + √3/22