Question

√(112) - √(80) / √(20) - √(28) = ?​

Answer 1

$\large\underline{\sf{Solution-}}$

We need to solve,

$\dfrac{\sqrt{112}-\sqrt{80}}{\sqrt{20}-\sqrt{28}}$

We can write the above mentioned expression as,

$\sf\longmapsto\dfrac{\sqrt{2\times2\times2\times2\times7}-\sqrt{2\times2\times2\times5}}{\sqrt{2\times2\times5}-\sqrt{2\times2\times7}}$

So,

$\sf\longmapsto\dfrac{\sqrt{4\times4\times7}-\sqrt{4\times4\times5}}{\sqrt{2^2\times5}-\sqrt{2^2\times7}}$

So,

$\sf\longmapsto\dfrac{\sqrt{4^2\times7}-\sqrt{4^2\times5}}{\sqrt{2^2\times5}-\sqrt{2^2\times7}}$

Hence,

$\sf\longmapsto\dfrac{4\sqrt{7}-4\sqrt{5}}{2\sqrt{5}-2\sqrt{7}}$

Taking common in respective terms,

$\sf\longmapsto\dfrac{4(\sqrt{7}-\sqrt{5})}{2(\sqrt{5}-\sqrt{7})}$

On cancelling 4 with 2,

$\sf\longmapsto\dfrac{4\!\!\!/^{\:2}(\sqrt{7}-\sqrt{5})}{2\!\!\!/(\sqrt{5}-\sqrt{7})}$

$\sf\longmapsto\dfrac{2(\sqrt{7}-\sqrt{5})}{(\sqrt{5}-\sqrt{7})}$

Taking (-) common,

$\sf\longmapsto\dfrac{-2(\sqrt{5}-\sqrt{7})}{(\sqrt{5}-\sqrt{7})}$

So, (√5 - √7) gets cut,

Hence,

$\longmapsto\bf\dfrac{\sqrt{112}-\sqrt{80}}{\sqrt{20}-\sqrt{28}}=-2$