Question

Simplify the rational expression by rationalizing the denominator. (1 point) Responses Image with alt text: start fraction start root 30 x end root over 2x end fraction Image with alt text: start fraction start root 10 x end root over 2x end fraction Image with alt text: start fraction start root 15 x end root over x end fraction

Answer 1

To simplify the given rational expression by rationalizing the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.

The given expression is:
$\frac{\sqrt{15x}}{x}$

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of $x$, which is $\sqrt{x}$:
$\frac{\sqrt{15x}}{x}\times \frac{\sqrt{x}}{\sqrt{x}}$
$=\frac{\sqrt{15x}\cdot \sqrt{x}}{x\cdot \sqrt{x}}$
$=\frac{\sqrt{15x^2}}{x\cdot \sqrt{x}}$
$=\frac{\sqrt{15}\cdot \sqrt{x^2}}{\sqrt{x}\cdot x}$
$=\frac{x\sqrt{15}}{x\sqrt{x}}$

Therefore, the simplified rational expression after rationalizing the denominator is $\frac{x\sqrt{15}}{x\sqrt{x}}$.