What is the simplified form of the following expression? Assume [tex]Y \neq 0[/tex]
[tex]\sqrt[3]{\frac{12 x^2}{16 y}}[/tex]
A. [tex]\frac{2\left(\sqrt[3]{6 x^2 y^2}\right)}{y}[/tex]
B. [tex]\frac{\sqrt[3]{12 x^2 y}}{2 y}[/tex]
C. [tex]\frac{x(\sqrt[3]{3 y})}{2 y}[/tex]
D. [tex]\sqrt[3]{6 x^2 y^2}[/tex]
Answer (2)
To simplify the expression 3 16 y 12 x 2 , we first simplify the fraction to 4 y 3 x 2 . Next, we rationalize the denominator, leading us to the final simplified expression 2 y 3 6 x 2 y 2 , which matches option B.
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Simplify the fraction inside the cube root: 16 y 12 x 2 = 4 y 3 x 2 .
Multiply the numerator and denominator inside the cube root by 2 y 2 to rationalize the denominator.
Simplify the cube root: 3 8 y 3 6 x 2 y 2 = 2 y 3 6 x 2 y 2 .
The simplified expression is 2 y 3 6 x 2 y 2 .
Explanation
Understanding the Problem We are given the expression 3 16 y 12 x 2 and the condition y = 0 . Our goal is to simplify this expression.
Simplifying the Fraction First, simplify the fraction inside the cube root: 16 y 12 x 2 = 4 y 3 x 2 So the expression becomes 3 4 y 3 x 2 .
Rationalizing the Denominator To rationalize the denominator inside the cube root, we want to make the denominator a perfect cube. We can achieve this by multiplying the numerator and denominator by 2 y 2 : 3 4 y 3 x 2 ⋅ 2 y 2 2 y 2 = 3 8 y 3 6 x 2 y 2 = 3 8 y 3 3 6 x 2 y 2 = 2 y 3 6 x 2 y 2
Final Answer Therefore, the simplified form of the given expression is 2 y 3 6 x 2 y 2
Examples
Imagine you are designing a container, and its volume is given by the expression 3 16 y 12 x 2 . Simplifying this expression helps you understand the relationship between the variables and optimize the container's dimensions for efficient storage or transportation. This kind of simplification is useful in engineering and design to make calculations easier and to better understand how different factors affect the outcome.
