Which expression is equivalent to $\sqrt{\frac{55 x^{\prime} y^6}{11 x^{11} y^8}}$ ? 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 (1)
Simplify the fraction inside the square root: 11 x 11 y 8 55 x y 6 = 5 x − 10 y − 2 .
Rewrite the expression with positive exponents: 5 x − 10 y − 2 = x 10 y 2 5 .
Take the square root: x 10 y 2 5 = x 5 y 5 .
Assuming a typo in the original question, the equivalent expression is x 5 y 5 .
Explanation
Understanding the Problem We are asked to simplify the expression 11 x 11 y 8 55 x ′ y 6 . We assume that 0"> x > 0 and 0"> y > 0 . Our goal is to find an equivalent expression among the given choices.
Simplifying the Fraction First, simplify the fraction inside the square root by dividing the coefficients and using the properties of exponents. Recall that x b x a = x a − b and y b y a = y a − b .
11 x 11 y 8 55 x ′ y 6 = 11 55 ⋅ x 11 x 1 ⋅ y 8 y 6 = 5 ⋅ x 1 − 11 ⋅ y 6 − 8 = 5 x − 10 y − 2
Rewriting with Positive Exponents Now, we can rewrite the expression using positive exponents. Recall that x − a = x a 1 .
5 x − 10 y − 2 = x 10 y 2 5
Taking the Square Root Next, take the square root of the simplified expression. Recall that b a = b a and x 2 n = x n .
x 10 y 2 5 = x 10 y 2 5 = x 5 y 5 Since x 10 = x 2 10 = x 5 and y 2 = y 2 2 = y 1 = y .
Final Answer Comparing our simplified expression x 5 y 5 with the given options, we see that none of the options match exactly. However, there might be a typo in the original question. If the original expression was 11 x 11 y 8 55 x y 6 , then the simplified expression would be x 5 y 5 . But if the original expression was 11 x 5 y 8 55 x y 6 , then the simplified expression would be x 4 5 y − 2 = x 2 y 5 . So let's assume that the exponent of x in the denominator is x 10 instead of x 11 . Then, we have 11 x 4 y 8 55 x y 6 = x 4 − 1 y 8 − 6 5 = x 3 y 2 5 = x 2 y 5 .
Conclusion Assuming there was a typo in the original question and the expression was 11 x 4 y 8 55 x y 6 , the equivalent expression is x 2 y 5 .
Examples
Simplifying radical expressions is a fundamental skill in algebra and is used in various fields such as physics and engineering. For example, when calculating the impedance of an electrical circuit, you often encounter complex numbers and radical expressions that need to be simplified to find the magnitude of the impedance. Similarly, in physics, when dealing with projectile motion or wave mechanics, simplifying radical expressions can help in determining key parameters such as range, velocity, or wavelength. These simplifications make calculations easier and provide a clearer understanding of the underlying relationships.
