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 square root of

a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b}

Accepted Answer

Let $\sqrt{(a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b})} = \sqrt{x} - \sqrt{y} + \sqrt{z} . . . (1)$

where $x, y, z > 0$

Squaring both sides, we get

$a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b} = (\sqrt{x} - \sqrt{y} + \sqrt{z})^{2}$

$\Rightarrow a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b} = x + y + z - 2 \sqrt{x y} - 2 \sqrt{y z} + 2 \sqrt{x z}$

Equating rational and irrational parts of both sides, we get

$x + y + z = a + 3 b + 4 . . . (i)$

$x z = 4 a . . . (ii)$

$y z = 3 a b . . . (iii)$

$x y = 12 b . . . (iv)$

Multiplying $(ii), (iii) & (iv)$ , we get

$x^{2} y^{2} z^{2} = 4 a \cdot 3 a b \cdot 4 \cdot 3 b = (12 a b)^{2}$

$\Rightarrow x y z = \pm 12 a b . . . (v)$

By $(ii), (iii), (iv) & (v)$ , we get $x = 4, y = 3 b, z = a$ which satisfy $(i)$ .

Hence, the required square root is

$[\sqrt{4} - \sqrt{3 b} + \sqrt{a}] = 2 + \sqrt{a} - \sqrt{3 b}$

Find the square root of $a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b}$

Important Questions on Surds and Imaginary Quantities

Learn and Prepare for any exam you want !

Find the square root of a+3b+4+4a-43b-23ab

Important Questions on Surds and Imaginary Quantities

Learn and Prepare for any exam you want !

Find the square root of $a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b}$