Hi can someone help me please simplify the following showing the working out step by step?
$$
\left({\sqrt{x} + \frac{1}{\sqrt{x}}}\right)^2 – \left({\sqrt{x} – \frac{1}{\sqrt{x}}}\right)^2
$$
I can't get the answer matching the text book but I'd also like to get an idea of the most idiomatic way to solve it in terms of steps.
What I attempt to do based on what I've learned so far is to:
- try and simplify the contents of the parens using conjugate and then LCM
- then square
- then handle the subtraction.
But my answer ends up incorrect.
So even steps just to simplify say the left hand term (without the squaring step) would be helpful.
My simplifying the left hand looks like:
Use Conjugate:
$$
\left(\sqrt{x} + \frac{(1)(\sqrt{x})}{(\sqrt{x})(\sqrt{x})}\right)^2
$$
$$
\left(\sqrt{x} + \frac{(\sqrt{x})}{x}\right)^2
$$
Use LCM:
$$
\left(\frac{(\sqrt{x})(x)}{x} + \frac{(\sqrt{x})}{x}\right)^2
$$
$$
\left(\frac{(x)(\sqrt{x}) + \sqrt{x}}{x}\right)^2
$$
Then I'm not sure next best step.
Best Answer
Using $a^2-b^2 = (a+b)(a-b)$ we get $$\left(\sqrt{x} + \frac{1}{\sqrt{x}} + \sqrt{x} - \frac{1}{\sqrt{x}}\right)\left(\sqrt{x}+\frac{1}{\sqrt{x}} - \sqrt{x}+\frac{1}{\sqrt{x}}\right) = \left(2\sqrt{x}\right)\left(\frac{2}{\sqrt{x}}\right) = 4$$ Hence, we get our answer as $4$.
Hope it helps.