Question

simplify 3√5-√7/3√3+√2 by rationalsing the denominator​

Answer 1

Question -

$Simplify \: \: \frac{3 \sqrt{5} - \sqrt{7} }{3 \sqrt{3} + \sqrt{2} } \: by \: \: \\ rationalising \: \: the \: \: denominator$

Solution -

$= \frac{3 \sqrt{5} - \sqrt{7} }{3 \sqrt{3} + \sqrt{2} }$

By rationalising the denominator we will get -

$= \frac{3 \sqrt{5} - \sqrt{7} }{3 \sqrt{ 3} + \sqrt{2} } \times \frac{3 \sqrt{3} - \sqrt{2} }{3 \sqrt{3} - \sqrt{2} }$

Using the identity (a+b) (a-b) =a²-b²

$= \frac{3 \sqrt{5} (3 \sqrt{3} - \sqrt{2} ) - \sqrt{7} (3 \sqrt{3} - \sqrt{2}) }{ ({3 \sqrt{3} )}^{2} - {( \sqrt{2}) }^{2} }$

$= \frac{9 \sqrt{15} - 3 \sqrt{10} - 3 \sqrt{21} + \sqrt{14} }{27 - 2}$

$= \frac{9 \sqrt{15} - 3 \sqrt{10} - 3 \sqrt{21} + \sqrt{14} }{25}$

Related Indentities -

$1) \sqrt{ab} = \sqrt{a} \sqrt{b}$

$2) \sqrt{ \frac{a}{b} } = \frac{ \sqrt{a} }{ \sqrt{b} }$

$3)( \sqrt{a} + \sqrt{b} )( \sqrt{a} - \sqrt{b}) = a - b$

$4)(a + \sqrt{b} )(a - \sqrt{b} ) = {a}^{2} - b$

$5)( \sqrt{a} + \sqrt{b} )( \sqrt{c} + \sqrt{d}) = \sqrt{ac} + \sqrt{ad} + \sqrt{bc} + \sqrt{bd}$

$6) {( \sqrt{a} + \sqrt{b}) }^{2} = a + 2 \sqrt{ab} + b$