Name: Rationalisation of Surds
Uploaded: 2019-07-19
Duration: 5 min 28 s
Description: When the denominator of an expression is a surd that can be reduced to an expression with a rational denominator, this process is called the rationalisation of a surd.

Question

Express as an equivalent fraction with rational denominator,

$\frac{2 \sqrt{a + 1}}{\sqrt{a - 1} - \sqrt{2 a} + \sqrt{a + 1}}$

Accepted Answer

Given expression is

$\frac{2 \sqrt{a + 1}}{\sqrt{a - 1} - \sqrt{2 a} + \sqrt{a + 1}}$

Multiplying and dividing by the conjugate of denominator i.e., $\{(\sqrt{a - 1} + \sqrt{a + 1}) + (\sqrt{2 a})\}$ , we get

$= [\frac{2 \sqrt{a + 1} \{(\sqrt{a - 1} + \sqrt{a - 1}) + (\sqrt{2 a})\}}{\{(\sqrt{a - 1} + \sqrt{a - 1}) - (\sqrt{2 a})\} \times \{(\sqrt{a - 1} + \sqrt{a - 1}) + (\sqrt{2 a})\}}]$

$= [\frac{2 \sqrt{a + 1} \{(\sqrt{a - 1} + \sqrt{a + 1}) + (\sqrt{2 a})\}}{\{{(\sqrt{a - 1} + \sqrt{a + 1})}^{2} - {(\sqrt{2 a})}^{2}\}}]$

$= [\frac{2 \sqrt{a^{2} - 1} + 2 (a + 1) + 2 \sqrt{2 a^{2} + 2 a)}}{a - 1 + a + 1 + 2 \sqrt{a^{2} - 1} - 2 a}]$

$= [\frac{\{\sqrt{a^{2} - 1} + (a + 1) + \sqrt{2 a^{2} + 2 a}\} \cdot \sqrt{a^{2} - 1}}{(\sqrt{a^{2} - 1} \cdot \sqrt{a^{2} - 1})}]$

$= [\frac{(a^{2} - 1) + (a + 1) \sqrt{a^{2} - 1} + \sqrt{(a + 1) (a - 1)} \cdot \sqrt{2 a (a + 1)}}{(a + 1) (a - 1)}]$

$= \frac{(a - 1) + \sqrt{a^{2} - 1} + \sqrt{2 a (a - 1)}}{a - 1} .$

Express as an equivalent fraction with rational denominator,

$\frac{2 \sqrt{a + 1}}{\sqrt{a - 1} - \sqrt{2 a} + \sqrt{a + 1}}$

Important Questions on Surds and Imaginary Quantities

Learn and Prepare for any exam you want !

Express as an equivalent fraction with rational denominator, 2a+1a-1-2a+a+1

Important Questions on Surds and Imaginary Quantities

Learn and Prepare for any exam you want !

Express as an equivalent fraction with rational denominator,

$\frac{2 \sqrt{a + 1}}{\sqrt{a - 1} - \sqrt{2 a} + \sqrt{a + 1}}$