Question

rationalize 1/√5+√3-√2​

Answer 1

Answer:

$\frac{1}{ \sqrt{5} } + \sqrt{3} + \sqrt{2 } \\ = \frac{1}{ \sqrt{5} } \times \frac{ \sqrt{5} }{ \sqrt{5} } + \sqrt{3 } + \sqrt{2} \\ = \frac{ \sqrt{5} }{5} + \sqrt{3} + \sqrt{2} \\ = \frac{ \sqrt{5} + 5 \sqrt{3} + 5 \sqrt{2} }{5}$

is the answer

Answer 2

Given :

Rationalize :

$\\ \bullet \bf \: \frac{1}{ \sqrt{5} + \sqrt{3} - \sqrt{2} }$

Formula :

• For rationalizing the equation we multiply in numerator and denominator by its conjugate.

$\\ \bullet \bf \: \frac{1}{a + b} \implies \frac{1}{a + b} \times \frac{a - b}{a - b} = \frac{a - b}{ {(a}^{2} - {b}^{2}) }$

$\\ \bullet \bf \: \frac{1}{a - b} \implies \ \: \frac{1}{a - b} \times \frac{a + b}{a + b} = \frac{a + b}{ {a}^{2} - {b}^{2} }$

Solution :

Given equation is :

$\\ \bullet \bf \frac{1}{ \sqrt{5} + \sqrt{3} - \sqrt{2} }$

• By using Formula :

•Multiplying $\bf{\sqrt{5}+\sqrt{3}- \sqrt{2}}$ in numerator and denominator we get,

$\\ \implies \sf \: \frac{1}{ \sqrt{5} + \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{5} + \sqrt{3} + \sqrt{2} }{ \sqrt{5} + \sqrt{3} + \sqrt{2} }$

$\\ \implies \sf \: \frac{ \sqrt{5} + \sqrt{3} + \sqrt{2} }{ {( \sqrt{5} + \sqrt{3}) }^{2} - {( \sqrt{2} )}^{2} }$

$\\ \implies \sf \: \frac{ \sqrt{5} + \sqrt{3} + \sqrt{2} }{ {( \sqrt{5}) }^{2} + {( \sqrt{3}) }^{2} + 2( \sqrt{5} )( \sqrt{3}) - 2 }$

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

$\\ \implies \sf \: \frac{ \sqrt{5} + \sqrt{3} + \sqrt{2} }{8 - 2 + 2 \sqrt{15} }$

$\\ \implies \sf \: \frac{ \sqrt{5} + \sqrt{3} + \sqrt{2} }{6 + 2 \sqrt{15} }$

• Multiplying $\bf{6-2 \sqrt{15} }$ in numerator and denominator we get,

$\\ \implies \sf \: \frac{ \sqrt{5} + \sqrt{3} + \sqrt{2} }{6 + 2 \sqrt{15} } \times \frac{6 - 2 \sqrt{15} }{6 - 2 \sqrt{15} }$

$\\ \implies \sf \: \frac{ \sqrt{5} (6 - 2 \sqrt{15} ) + \sqrt{3} (6 - 2 \sqrt{15}) + \sqrt{2} (6 - 2 \sqrt{15} )}{ {(6)}^{2} - {(2 \sqrt{15} )}^{2} }$

$\\ \implies \sf \: \frac{6 \sqrt{5} - 2 \sqrt{15 \times 5} + 6 \sqrt{3} - 2 \sqrt{45} + 6 \sqrt{2} - 2 \sqrt{30} }{36 - 60}$

$\\ \implies \sf \: \frac{6 \sqrt{5} - 10\sqrt{3} + 6 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{2} - 2 \sqrt{30} }{ - 24}$

$\\ \implies \sf \: \frac{ - 4 \sqrt{3} + 6 \sqrt{2} - 2 \sqrt{30} }{ - 24}$

$\\ \implies \sf \: \frac{ \cancel- (4 \sqrt{3} - 6 \sqrt{2} + 2 \sqrt{30} )}{ \cancel- 24}$

$\implies \boxed{ \bf{ \frac{4 \sqrt{3} - 6\sqrt{2} + 2 \sqrt{30} }{24} }}$