Which expression is equivalent to $\sqrt{\frac{55 x^7 y^6}{11 x^{11} y^8}}$ ? Assume $x>0$ and $y>0$
A. $\frac{x^2 \sqrt{5}}{y}$
B. $\frac{y \sqrt{5}}{x^2}$
C. $\frac{\sqrt{5}}{x^2 y}$
D. $\frac{x \sqrt{5}}{y}$
Answer (2)
Simplify the fraction inside the square root: 11 x 11 y 8 55 x 7 y 6 = 5 x − 4 y − 2 .
Substitute the simplified fraction back into the square root: 5 x − 4 y − 2 .
Rewrite the expression using positive exponents: x 4 y 2 5 .
Simplify the square root: x 2 y 5 . The final answer is x 2 y 5 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to 11 x 11 y 8 55 x 7 y 6 , assuming that 0"> x > 0 and 0"> y > 0 .
Simplifying the Fraction First, we simplify the fraction inside the square root: 11 x 11 y 8 55 x 7 y 6 = 11 55 ⋅ x 11 x 7 ⋅ y 8 y 6 = 5 ⋅ x 7 − 11 ⋅ y 6 − 8 = 5 x − 4 y − 2 .
Substituting Back Now we substitute the simplified fraction back into the square root: 5 x − 4 y − 2 .
Rewriting with Positive Exponents Next, we rewrite the expression using positive exponents: x 4 y 2 5 .
Simplifying the Square Root Finally, we simplify the square root: x 4 y 2 5 = x 2 y 5 .
Final Answer Therefore, the expression equivalent to 11 x 11 y 8 55 x 7 y 6 is x 2 y 5 .
Examples
Imagine you are calculating the dimensions of a rectangular garden plot. You start with a complex formula involving the ratio of the areas and side lengths, similar to the expression we just simplified. By simplifying such expressions, you can more easily determine the actual dimensions needed for your garden, making your planning and calculations much more straightforward. This is also applicable in physics, where simplifying complex formulas helps in understanding relationships between different physical quantities.
The expression 11 x 11 y 8 55 x 7 y 6 simplifies to x 2 y 5 . Therefore, the chosen answer is (C) x 2 y 5 .
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