# Thread: What is the Square of the Square Root of a Polynomial

1. ## What is the Square of the Square Root of a Polynomial

Example: $\displaystyle (\sqrt{1-x^2})^2$

Inuitively, I would say it's simply $\displaystyle 1-x^2$. Alternatively $\displaystyle (\sqrt{1-x^2})(\sqrt{1-x^2})$ appears to give $\displaystyle \sqrt{1-2x^2+4x^2}$ which also does not seem correct.

The question arises from this problem:

$\displaystyle \sqrt{1+(\frac{x}{\sqrt{1-x^2}})^2}$ where I constantly get 1 by stating $\displaystyle \sqrt{1-x^2} = 1-x^2$. The correct answer, according to my text book anyway, is $\displaystyle \frac{1}{\sqrt{1-x^2}}$.

Thanks -

2. Nevermind, I completely forgot that I needed to multiply the denominator by $\displaystyle \sqrt{1-x^2}$ to keep the expression equal. The numerator as I simplified earlier, is 1 and there is nothing to simplify in the denominator $\displaystyle => \frac{1}{\sqrt{1-x^2}}$

3. Originally Posted by TaylorM0192
Example: $\displaystyle (\sqrt{1-x^2})^2$

Inuitively, I would say it's simply $\displaystyle 1-x^2$. Alternatively $\displaystyle (\sqrt{1-x^2})(\sqrt{1-x^2})$ appears to give $\displaystyle \sqrt{1-2x^2+4x^2}$ which also does not seem correct.

The question arises from this problem:

$\displaystyle \sqrt{1+(\frac{x}{\sqrt{1-x^2}})^2}$ where I constantly get 1 by stating $\displaystyle \sqrt{1-x^2} = 1-x^2$. The correct answer, according to my text book anyway, is $\displaystyle \frac{1}{\sqrt{1-x^2}}$.

Thanks -
Help me out:
So you distributed the exponents to $\displaystyle \frac {x}{\sqrt{1-x^2}}$ , how would you simplify ... ohh I see.

So it simplifies to ..
$\displaystyle \sqrt {\frac {1}{1-x^2} + \frac {x^2}{1-x^2}}$

right? If so, then where did the $\displaystyle x^2$ in the numerator go?

4. ## algebraic calculation

√(1+((x/(√(1-x²))))²)
=√(1+(((x²)/(1-x²))))
=√(((1-x²+x²)/(1-x²)))
=√((1/(1-x²)))
=(1/(√(1-x²)))