The simplified expression by rationalizing the denominator is (C)[tex] \frac{4 \sqrt{15x} }{15x} [/tex].
First we must simplify the expression:
[tex] \frac{3 \sqrt{160} }{\sqrt{1350x} } = \frac{12 \sqrt{10} }{15 \sqrt{6x} } = \frac{4 \sqrt{10} }{5 \sqrt{6x} }[/tex]
Then we factor the rational parts and cancel it out:
[tex]\frac{4 \sqrt{2} \sqrt{5} }{5 \sqrt{2} \sqrt{3x} } = \frac{4\sqrt{5} }{5\sqrt{3x} } [/tex]
Then we rationalize the expression:
[tex]\frac{4\sqrt{5} }{5\sqrt{3x} } * \frac{\sqrt{3x} }{\sqrt{3x} } = \frac{4 \sqrt{15x} }{5*3x} = \frac{4 \sqrt{15x} }{15x}[/tex]
Finally, the simplified expression by rationalizing the denominator is (C)[tex] \frac{4 \sqrt{15x} }{15x} [/tex].
