- Simplify the coefficients and exponents within the square root: \(\frac{55 x^7 y^6}{11 x^{11} y^8} = \frac{5}{x^4 y^2}\).
- Extract the square root of the simplified expression: \(\sqrt{\frac{5}{x^4 y^2}} = \frac{\sqrt{5}}{x^2 y}\).
- Compare the result with the given choices to find the equivalent expression: \(\frac{\sqrt{5}}{x^2 y}\).
- Final answer: \(\boxed{\frac{\sqrt{5}}{x^2 y}}\)
Explanation
- Analyzing the Expression
We are given the expression \(\sqrt{\frac{55 x^7 y^6}{11 x^{11} y^8}}\) with the conditions that \(x > 0\) and \(y > 0\). Our goal is to simplify this expression by applying algebraic laws for fractions, exponents, and square roots, and then match the result with the provided options.
- Simplifying the Fraction Inside the Square Root
First, we simplify the fraction inside the square root. We divide the coefficients and the variables separately.
For the coefficients:
\[\frac{55}{11} = 5\]
For the variable \(x\):
\[\frac{x^7}{x^{11}} = x^{7-11} = x^{-4} = \frac{1}{x^4}\]
For the variable \(y\):
\[\frac{y^6}{y^8} = y^{6-8} = y^{-2} = \frac{1}{y^2}\]
Combining these, the expression inside the square root becomes \(\frac{5}{x^4 y^2}\).
- Evaluating the Square Root
Now we apply the square root to the simplified fraction:
\[\sqrt{\frac{5}{x^4 y^2}} = \frac{\sqrt{5}}{\sqrt{x^4 y^2}}\]
Using the property of square roots \(\sqrt{a^2 b^2} = ab\) for positive \(a, b\), we have \(\sqrt{x^4 y^2} = \sqrt{(x^2)^2 \cdot y^2} = x^2 y\).
Thus, the expression simplifies to \(\frac{\sqrt{5}}{x^2 y}\).
- Conclusion and Final Selection
Comparing our simplified result \(\frac{\sqrt{5}}{x^2 y}\) with the given options, we identify that it matches the third option. Therefore, the equivalent expression is \(\frac{\sqrt{5}}{x^2 y}\).
Examples
Algebraic simplification is a fundamental skill used in many scientific fields. For example, in physics, when calculating the intensity of light or sound at a certain distance, expressions often involve square roots and ratios of variables. Simplifying these expressions allows engineers and scientists to compute results more quickly and accurately, whether they are designing communication systems, acoustics for a concert hall, or analyzing sensor data from equipment. Proficiency in handling these operations ensures that complex real-world models can be broken down into manageable and interpretable calculations.