What is the following product? Assume [tex]$x \geq 0$[/tex] and [tex]$y \geq 0$[/tex].
[tex]$\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$[/tex]
A. [tex]$3 x^5 y \sqrt{3 x y}$[/tex]
B. [tex]$10 x^5 y \sqrt{6 x y}$[/tex]
C. [tex]$3 x^3 y \sqrt{3 x^2 y^2}$[/tex]
D. [tex]$10 x^3 y \sqrt{6 x^2 y^2}$[/tex]
Answer (2)
The product 5 x 8 y 2 ⋅ 10 x 3 ⋅ 12 y simplifies to 10 x 5 y 6 x y . Therefore, the correct answer is option B . This process involves combining square roots, factoring out perfect squares, and simplifying the expression step by step.
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Combine the square roots: 5 x 8 y 2 ⋅ 10 x 3 ⋅ 12 y = 600 x 11 y 3 .
Factor out perfect squares: 600 x 11 y 3 = 100 ⋅ 6 ⋅ x 10 ⋅ x ⋅ y 2 ⋅ y .
Take the square root of perfect squares: 10 x 5 y 6 x y .
The simplified expression is 10 x 5 y 6 x y .
Explanation
Understanding the Problem We are given the expression 5 x 8 y 2 ⋅ 10 x 3 ⋅ 12 y with the conditions x ≥ 0 and y ≥ 0 . Our goal is to simplify this expression.
Combining Square Roots First, we combine the square roots into a single square root: 5 x 8 y 2 ⋅ 10 x 3 ⋅ 12 y = 5 x 8 y 2 ⋅ 10 x 3 ⋅ 12 y = 600 x 11 y 3 .
Factoring Perfect Squares Next, we simplify the square root by factoring out perfect squares: 600 x 11 y 3 = 100 ⋅ 6 ⋅ x 10 ⋅ x ⋅ y 2 ⋅ y = 1 0 2 ⋅ 6 ⋅ ( x 5 ) 2 ⋅ x ⋅ y 2 ⋅ y .
Taking Square Roots Now, we take the square root of the perfect squares: 1 0 2 ⋅ 6 ⋅ ( x 5 ) 2 ⋅ x ⋅ y 2 ⋅ y = 10 x 5 y 6 x y .
Final Answer Finally, we compare the simplified expression with the given options and choose the correct one. The simplified expression is 10 x 5 y 6 x y .
Examples
Square roots are used in many fields, such as physics and engineering, to calculate distances, areas, and volumes. For example, when calculating the speed of an object falling from a certain height, we use square roots. Also, in construction, square roots are used to determine the length of the diagonal of a square room or building. Understanding how to simplify expressions with square roots helps in these calculations.
