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
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 \( \frac{x \sqrt{15}}{x \sqrt{x}} \).