Question

Simplify √2/(√6-√2)-√3/(√6+√2)

Answer 1

Step-by-step explanation:

on taking lcm u can solve it

Answer 2

Answer:

The simplification of the above expression is

$\frac{ \sqrt{2} }{ \sqrt{6} - \sqrt{2} } - \frac{ \sqrt{3} }{ \sqrt{6} + \sqrt{2} } = \frac{2 \sqrt{3} + 2 + 3 \sqrt{2} - \sqrt{6} }{4}$

Step-by-step explanation:

The given expression in irrational terms is mentioned as below

$\frac{ \sqrt{2} }{ \sqrt{6} - \sqrt{2} } - \frac{ \sqrt{3} }{ \sqrt{6} + \sqrt{2} }$

As we can observe the denominators as a sum of two irrational numbers.To convert them into a proper number we shall Rationalize them.

A number can be rationalised in the following way by multiplying amd dividing it with its Rational Factor.

$\frac{1}{ \sqrt{a} + \sqrt{b} } =\frac{1}{ \sqrt{a} + \sqrt{b} } \times \frac{ \sqrt{a} - \sqrt{b} }{ \sqrt{a} - \sqrt{b} } \\ = \frac{ \sqrt{a} - \sqrt{b} }{a - b}$

Doing the rationalization for the given expression as follows,we get

$\frac{ \sqrt{2} }{ \sqrt{6} - \sqrt{2} } \times \frac{ \sqrt{6} + \sqrt{2} }{ \sqrt{6} + \sqrt{2} } \\ = \frac{ \sqrt{2} ( \sqrt{6} + \sqrt{2}) }{ { \sqrt{6} }^{2} - { \sqrt{2} }^{2} } = \frac{ \sqrt{12} + 2}{4} \\ = \frac{2( \sqrt{3} + 1)}{4}$

Similarly,doing it for the second term we get

$\frac{ \sqrt{3} }{ \sqrt{6} + \sqrt{2} } \times \frac{ \sqrt{6} - \sqrt{2} }{ \sqrt{6} - \sqrt{2} } \\ = \frac{ \sqrt{3} ( \sqrt{6} - \sqrt{2} )}{4} \\ = \frac{ \sqrt{18} - \sqrt{6} }{4}$

Adding both the terms we get

$\frac{2 \sqrt{3} + 2 }{4} + \frac{ \sqrt{18} - \sqrt{6} }{4} \\ = \frac{2 \sqrt{3} + 2 + 3 \sqrt{2} - \sqrt{6} }{4}$

Hence,the simpification of the above expression is found as

$\frac{ \sqrt{2} }{ \sqrt{6} - \sqrt{2} } - \frac{ \sqrt{3} }{ \sqrt{6} + \sqrt{2} } = \frac{2 \sqrt{3} + 2 + 3 \sqrt{2} - \sqrt{6} }{4}$

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