Question

Find the square root in the form of binomial surd:
$55-14\sqrt{6}$.

Answer 1

Square root of $55-14\sqrt{6}$:

Let $\sqrt{55-14\sqrt{6}}=\sqrt{a}-\sqrt{b}$.

Squaring of both sides,

$⟹$ $(\sqrt{(55-14\sqrt{6})}{)}^2=(\sqrt{a}-\sqrt{b}{)}^2$

$⟹$ $55-14\sqrt{6}$ $=a+b-2\sqrt{ab}$

$⟹$ $55-2\times 7\sqrt{6}$ $=a+b-2\sqrt{ab}$

$⟹$ $55-2\sqrt{7\times 7\times 6}$ $=a+b-2\sqrt{ab}$.

So $-2\sqrt{ab}=-2\sqrt{49\times 6}$.

Here, $a=49,b=6$.

Therefore,

Square root of $55-14\sqrt{6}$ is $(\sqrt{49}-\sqrt{6})$ $=(7-\sqrt{6})$.