# How does subtracting an exponent from an exponent return a greater value?

I'm just stuck on a problem where I have to simplify an expression. This is the expression:

$$\sqrt{x^2\:-\:\left(\frac{x}{2}\right)^2}$$

The textbook has the answer as:

$$\frac{\sqrt{3}x}{2}$$

I have no idea how to get to that. I've tried to figure it out for the past hour and I have tried online solvers but I can't understand what the steps are. Any help would be very much appreciated.

Thanks!

• What have you tried so far? Jan 12, 2021 at 2:21
• $x^2-\left(\dfrac x2\right)^2=\dfrac{3x^2}4$ Jan 12, 2021 at 2:28
• A little bit picky, but the text-book is wrong. The expression actually simplifies to $\frac {|x|\sqrt3}{2}$ Jan 12, 2021 at 2:36
• @DougM Not picky at all, I would say. Jan 12, 2021 at 2:40
• J.W. I understand that but where do you get the 3 from? Jan 12, 2021 at 2:49

You solution assumes that $$x\ge 0$$.
\begin{align} x^2-(\frac{x}{2})^2 &=x^2-\frac{x^2}{4}\\ &=1\cdot x^2-\frac{1}{4}\cdot x^2\\ &=(1-\frac{1}{4})x^2= \frac34 x^2 \end{align}
So $$\sqrt{x^2-(\frac{x}{2})^2}=\sqrt{\frac34 x^2} =\frac{\sqrt{3}}{\sqrt{4}}\sqrt{x^2}=\frac{\sqrt{3}}{2}x\quad x\ge 0$$