Question

rationalize the denominator of \\(\frac{5}{\sqrt{5x}}\\). assume that all variables represent positive real numbers.
\\(\frac{5}{\sqrt{5x}} = \square\\) (type an exact answer, using radicals as needed.)

Explanation:

Step1: Multiply numerator and denominator by $\sqrt{5x}$

To rationalize the denominator, we multiply the fraction $\frac{5}{\sqrt{5x}}$ by $\frac{\sqrt{5x}}{\sqrt{5x}}$ (since $\frac{\sqrt{5x}}{\sqrt{5x}} = 1$ and multiplying by 1 doesn't change the value of the fraction). So we get $\frac{5\times\sqrt{5x}}{\sqrt{5x}\times\sqrt{5x}}$.

Step2: Simplify the denominator

We know that $\sqrt{a}\times\sqrt{a}=a$ for $a\geq0$. So here, $\sqrt{5x}\times\sqrt{5x} = 5x$. The numerator is $5\sqrt{5x}$. So the fraction becomes $\frac{5\sqrt{5x}}{5x}$.

Step3: Cancel out the common factor

We can cancel out the common factor of 5 in the numerator and the denominator. So $\frac{5\sqrt{5x}}{5x}=\frac{\sqrt{5x}}{x}$.

Answer:

$\frac{\sqrt{5x}}{x}$