Question
Use the quotient rule to simplify. Assume that all variables represent positive real numbers. \( \sqrt{\frac{x^{2} y}{225}} \) \( \sqrt{\frac{x^{2} y}{225}}=\square \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)
Ask by Murray Reid. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{x \sqrt{y}}{15}
\]
Solution
We start with the expression
\[
\sqrt{\frac{x^2 y}{225}}
\]
and apply the quotient rule for square roots, which states that
\[
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}},
\]
for \( a, b > 0 \). Thus, we have
\[
\sqrt{\frac{x^2 y}{225}} = \frac{\sqrt{x^2 y}}{\sqrt{225}}.
\]
Next, compute the denominator:
\[
\sqrt{225} = 15.
\]
Now, simplify the numerator. Since \( x \) is positive, we know that
\[
\sqrt{x^2} = x.
\]
Thus,
\[
\sqrt{x^2 y} = \sqrt{x^2} \sqrt{y} = x \sqrt{y}.
\]
Putting these together, our expression becomes:
\[
\frac{x \sqrt{y}}{15}.
\]
This is the simplified form of the original expression.
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Bonus Knowledge
To simplify \( \sqrt{\frac{x^{2} y}{225}} \), we can apply the property of square roots that states \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). So, we rewrite it as: \[ \sqrt{\frac{x^{2} y}{225}} = \frac{\sqrt{x^{2} y}}{\sqrt{225}}. \] Now, we simplify each part: - \( \sqrt{225} = 15 \). - \( \sqrt{x^2 y} = \sqrt{x^2} \cdot \sqrt{y} = x \sqrt{y} \) since \( x \) is positive. Combining these, we have: \[ \frac{x \sqrt{y}}{15}. \] Thus, the simplified form is \[ \frac{x \sqrt{y}}{15}. \]