Questions: Simplify using the quotient rule for square roots. Assume that x>0. (sqrt(375 x^6))/(sqrt(5 x))

Solution Steps

To simplify the given expression using the quotient rule for square roots, we can combine the square roots into a single square root and then simplify the expression inside the square root. The quotient rule for square roots states that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).

1. Combine the square roots into a single square root: \(\frac{\sqrt{375 x^6}}{\sqrt{5 x}} = \sqrt{\frac{375 x^6}{5 x}}\).
2. Simplify the expression inside the square root: \(\frac{375 x^6}{5 x} = 75 x^5\).
3. Take the square root of the simplified expression: \(\sqrt{75 x^5}\).

Using the quotient rule for square roots, we combine the square roots into a single square root: \[ \frac{\sqrt{375 x^6}}{\sqrt{5 x}} = \sqrt{\frac{375 x^6}{5 x}} \]

Simplify the expression inside the square root: \[ \frac{375 x^6}{5 x} = 75 x^5 \]

Take the square root of the simplified expression: \[ \sqrt{75 x^5} \]

Further simplify the square root: \[ \sqrt{75 x^5} = \sqrt{75} \cdot \sqrt{x^5} = \sqrt{25 \cdot 3} \cdot \sqrt{x^4 \cdot x} = 5 \sqrt{3} \cdot x^2 \cdot \sqrt{x} \]

Final Answer