What is $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$ in simplest form? Assume $x \geq 0$ and $y \geq 0$.
A. $5 \sqrt{x y}$
B. $25 \sqrt{x y}$
C. $\sqrt{5 x y}$
D. $5 x y \sqrt{x y}$
Answer (2)
The expression x y 25 x 2 y 2 simplifies to 5 x y . The answer corresponds to option A. Thus, the final answer is 5 x y .
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Rewrite the expression using the properties of square roots: x y 25 x 2 y 2 = x y 5 x y .
Rationalize the denominator by multiplying the numerator and denominator by x y : x y 5 x y = x y 5 x y x y .
Cancel the common factor x y from the numerator and the denominator: x y 5 x y x y = 5 x y .
The simplified expression is: 5 x y .
Explanation
Understanding the Problem We are asked to simplify the expression x y 25 x 2 y 2 , assuming that x ≥ 0 and y ≥ 0 . This means that x and y are non-negative real numbers.
Simplifying the Expression Since x ≥ 0 and y ≥ 0 , we have x 2 = ∣ x ∣ = x and y 2 = ∣ y ∣ = y . Thus, we can rewrite the expression as follows: x y 25 x 2 y 2 = x y 25 x 2 y 2 = x y 5 x y
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by x y :
x y 5 x y = x y ⋅ x y 5 x y ⋅ x y = x y 5 x y x y
Final Simplification Now, we can cancel the common factor x y from the numerator and the denominator: x y 5 x y x y = 5 x y Thus, the simplified expression is 5 x y .
Examples
Imagine you are calculating the speed of a wave traveling through a medium, where the wave's energy is proportional to the square root of the product of two variables, x and y . Simplifying expressions like x y 25 x 2 y 2 helps in determining the wave's speed in terms of x y , making it easier to analyze and predict the wave's behavior. This type of simplification is useful in physics and engineering when dealing with wave phenomena, signal processing, and other related fields.
