How does \sqrt{1+((x^2)/(4-x^2))} simplify to 2 times\sqrt{1/(4-x^2)}?

[Waggattack]

1.I can't figure out how the $$\sqrt{1+((x^2)/(4-x^2))}$$ simplifies to 2 times$$\sqrt{1/(4-x^2)}$$


I have tried rewriting it in different ways, but I can't see how it simplifies. $$\sqrt{x^2 + 1/4-x^2}$$

 

[w3390]

The first thing to do is find a common denominator. Then you will be able to zero out some terms. Then, using the property of a square root, the square root of a fraction is the same as the square root of the numerator over the square root of the denominator. This will give you the answer.

 

[PhaseShifter]

$$\sqrt{1+((x^2)/(4-x^2))}$$ simplifies to 2 times $$\sqrt{1/(4-x^2)}$$

It may help to rewrite these in a form where you don't need the parentheses.

$$\sqrt{1+{{x^2}\over{4-x^2}}}$$ simplifies to $$2\sqrt{{{1}\over{4-x^2}}}$$
Does that make it easier?

 

[njama]

Hint: 1 in the square root, $$1=\frac{4-x^2}{4-x^2}$$

 

[HallsofIvy]

In other words, write
$$1+\frac{x^2}{4-x^2}$$
as
$$\frac{4- x^2}{4- x^2}+ \frac{x^2}{4- x^2}$$
and add the fractions.