I know this is a very basic question, but could someone please mathematically explain, why this is true:
$\sqrt{x} \cdot \frac{1}{x} = \frac{1}{\sqrt{x}}$
Wolfram|Alpha can confirm this.
fractions
I know this is a very basic question, but could someone please mathematically explain, why this is true:
$\sqrt{x} \cdot \frac{1}{x} = \frac{1}{\sqrt{x}}$
Wolfram|Alpha can confirm this.
Best Answer
Let us suppose that $x$ is positive. If it is not, our expression is not defined. Note that essentially by definition, $$x=\sqrt{x}\sqrt{x}.$$
It follows that $$\frac{\sqrt{x}}{x}=\frac{\sqrt{x}}{\sqrt{x}\sqrt{x}}=\frac{1}{\sqrt{x}}.$$ (For the last step, we divided top and bottom by $\sqrt{x}$.)