Question

Rationalise the Denominator
$\frac{1}{\sqrt{2} + \sqrt{3} }$
​

Answer 1

Step-by-step explanation:

Given:-

$\frac{1}{ \sqrt{2} + \sqrt{3} }$

What to do:-

To Rationalise the Denominator.

Solution:-

Let's solve the problem,

We have

$\frac{1}{ \sqrt{2} + \sqrt{3} }$

The denominator is √2+√3. Multiplying the numerator and denomination by √2-√3, we get

$⟹ \frac{1}{ \sqrt{2} + \sqrt{3} } \times \frac{ \sqrt{2} - \sqrt{3} }{ \sqrt{2} - \sqrt{3} }$

$⟹ \frac{ \sqrt{2} - \sqrt{3} }{( \sqrt{2} + \sqrt{3})( \sqrt{2} - \sqrt{3} ) }$

⬤ Applying Algebraic Identity

(a+b)(a-b) = a² - b² to the denominator

We get,

$⟹ \frac{ \sqrt{2} - \sqrt{3} }{( \sqrt{2} {)}^{2} - ( \sqrt{3} {)}^{2} }$

$⟹ \frac{ \sqrt{2} - \sqrt{3} }{2 - 3}$

$⟹ \frac{ \sqrt{2} - \sqrt{3} }{1}$

$⟹ \sqrt{2} - \sqrt{3} \: \: \: \tt{Ans.}$

Hence, the denominator is rationalised.

• I hope it's help you...☺

Know more Algebraic Identities:-

(a+ b)² = a² + b² + 2ab

( a - b )² = a² + b² - 2ab

( a + b )² + ( a - b)² = 2a² + 2b²

( a + b )² - ( a - b)² = 4ab

( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

a² + b² = ( a + b)² - 2ab

(a + b )³ = a³ + b³ + 3ab ( a + b)

( a - b)³ = a³ - b³ - 3ab ( a - b)

If a + b + c = 0 then a³ + b³ + c³ = 3abc