Question
Simplify \( \frac{\sqrt{x^{3} y^{2}}}{\sqrt{9 x^{7}}} \) \( \frac{y^{2}}{3 x} \) \( \frac{y^{2}}{3 x^{2}} \) \( \frac{y}{3 x^{2}} \)
Ask by Huff Johnston. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \( \frac{y}{3x^{2}} \).
Solution
We want to simplify
√(x³y²) ÷ √(9x⁷).
Step 1. Write the expression as one square root:
√(x³y²) / √(9x⁷) = √[(x³y²) / (9x⁷)].
Step 2. Simplify inside the square root. Notice that x³/x⁷ = 1/x⁴:
(x³y²)/(9x⁷) = y²/(9x⁴).
Step 3. Split the square root over the numerator and denominator:
√[y²/(9x⁴)] = √(y²) / √(9x⁴).
Step 4. Evaluate each square root. Assuming x and y are nonnegative,
√(y²) = y,
√(9x⁴) = 3x².
Thus, the expression simplifies to
y/(3x²).
So the final simplified form is y/(3x²).
Comparing with the given alternatives, the correct answer is y/(3 x²).
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Simplify this solution Beyond the Answer
To simplify \( \frac{\sqrt{x^{3} y^{2}}}{\sqrt{9 x^{7}}} \), we can combine the square roots: \[ \frac{\sqrt{x^{3} y^{2}}}{\sqrt{9 x^{7}}} = \sqrt{\frac{x^{3} y^{2}}{9 x^{7}}} \] Next, simplify the fraction inside the square root: \[ \frac{x^{3} y^{2}}{9 x^{7}} = \frac{y^{2}}{9} \cdot \frac{x^{3}}{x^{7}} = \frac{y^{2}}{9} \cdot \frac{1}{x^{4}} = \frac{y^{2}}{9 x^{4}} \] Now, taking the square root: \[ \sqrt{\frac{y^{2}}{9 x^{4}}} = \frac{\sqrt{y^{2}}}{\sqrt{9} \cdot \sqrt{x^{4}}} = \frac{y}{3 x^{2}} \] Thus, the simplified form of \( \frac{\sqrt{x^{3} y^{2}}}{\sqrt{9 x^{7}}} \) is \[ \frac{y}{3 x^{2}} \] So, the final answer is \( \frac{y}{3 x^{2}} \).
