Question

how to proceed the step....and in general how to remove the under root number/what's the basic thing that we should follow to do...when we have square +real number or etc... anything else

Answer 1

After that step,

$RS = \sqrt{4 {a}^{2} + 4 {b}^{2} } \\ \\ RS = \sqrt{4( {a}^{2} + {b}^{2} )}$

$RS = \sqrt{ {2}^{2} \times ( {a}^{2} + {b}^{2}) } \\ \\ RS = \sqrt{ {2}^{2}} \times \sqrt{ ( {a}^{2} + {b}^{2}) } \\ \\ RS = 2 \sqrt{ {a}^{2} + {b}^{2} }$

We can simplify it hoti that point.

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To remove square root, you can normally do two things:

1. Easiest thing to do is keep the entities under square root on side of equation and other entities on the other side of the equation and SQUARE BOTH SIDES.

for example, if you are given
$x = \sqrt{x + 2} - 6$
Take the things with square root to one side and all other things to another side
$x + 6 = \sqrt{x + 2}$
Now square both sides to remove square
$({x + 6})^{2} =( { \sqrt{x + 2} })^{2} \\ \\ {{x}^{2} + 12x + 36} = x + 2 \\ \\ {x}^{2} + 11x + 34 = 0$
Now there is no square root.

2. Another way is to convert the things inside under root to perfect squares. But not everything can be converted to perfect square. For example, when the RS is simplified, 4 could be converted to perfect square; so I took √4 outside. √4 is 2. so 2 is outside of square root. But (a^2 + b^2) can't be converted to a perfect square. So it is still under root. And it will stay that way.

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When you have real numbers and variables under square root, first take common as much as possible. Then try to convert them to perfect squares. You can take the roots of perfect squares and bring it outside of the square root. But if it is not a perfect square, then it will stay inside under root.