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

Find the value of

\frac{\sqrt{26 - 15 \sqrt{3}}}{5 \sqrt{2} - \sqrt{38 + 5 \sqrt{3}}}

.

Accepted Answer

Given, $\frac{\sqrt{26 - 15 \sqrt{3}}}{5 \sqrt{2} - \sqrt{38 + 5 \sqrt{3}}}$

First we will find the value of $\sqrt{(26 - 15 \sqrt{3})}$ .

Let, $\sqrt{(26 - 15 \sqrt{3})} = \sqrt{x} - \sqrt{y} . . . (i)$ , where $x, y > 0$

Squaring both sides, we get

$26 - 15 \sqrt{3} = (\sqrt{x} - \sqrt{y})^{2}$

$\Rightarrow 26 - 15 \sqrt{3} = (x + y - 2 \sqrt{x y})$

Equating rational and irrational parts, we get

$x + y = 26 . . . (ii)$

And, $2 \sqrt{x y} = 15 \sqrt{3}$

$\Rightarrow 4 x y = 675 . . . (iii)$

Solving $(ii)$ and $(iii)$ for $x$ and $y$ , we get

$x = \frac{27}{2}, y = \frac{25}{2}$

$∴ \sqrt{(26 - 15 \sqrt{3})} = \sqrt{\frac{27}{2}} - \sqrt{\frac{25}{2}}$

$= 3 \sqrt{\frac{3}{2}} - \frac{5}{\sqrt{2}} . . . (iv)$

Again, $38 + 5 \sqrt{3} = \frac{1}{2} (76 + 2 \sqrt{75})$

$= \frac{1}{2} (75 + 1 + 2 \times \sqrt{75} \times 1)$

$= \frac{1}{2} {(\sqrt{75} + 1)}^{2}$

Therefore, required value of

$\sqrt{(38 + 5 \sqrt{3})} = \sqrt{\{\frac{1}{2} {(\sqrt{75} + 1)}^{2}\}}$

$= \frac{(\sqrt{75} + 1)}{\sqrt{2}}$

$= (\frac{5 \sqrt{3}}{\sqrt{2}} + \frac{1}{\sqrt{2}}) . . . (v)$

Now, $\frac{\sqrt{(26 - 15 \sqrt{3})}}{5 \sqrt{2} - \sqrt{(38 + 5 \sqrt{3})}} = \{\frac{3 \sqrt{\frac{3}{2}} - \frac{5}{\sqrt{2}}}{5 \sqrt{2} - (5 \sqrt{\frac{3}{2}} + \frac{1}{\sqrt{2}})}\}$

$= \frac{(3 \sqrt{3} - 5)}{5 \sqrt{2} \times \sqrt{2} - 5 \sqrt{3} - 1}$

$= \frac{(3 \sqrt{3} - 5) (9 + 5 \sqrt{3})}{(9 - 5 \sqrt{3}) (9 + 5 \sqrt{3})}$

$= \frac{27 \sqrt{3} - 25 \sqrt{3} - 45 + 45}{81 - 75}$

$= \frac{2 \sqrt{3}}{6} = \frac{\sqrt{3}}{3}$

H S Hall and S R Knight Solutions for Chapter: Surds and Imaginary Quantities, Exercise 1: EXAMPLES. VIII. a.

H S Hall Mathematics Solutions for Exercise - H S Hall and S R Knight Solutions for Chapter: Surds and Imaginary Quantities, Exercise 1: EXAMPLES. VIII. a.

Questions from H S Hall and S R Knight Solutions for Chapter: Surds and Imaginary Quantities, Exercise 1: EXAMPLES. VIII. a. with Hints & Solutions

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