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]
Answer (2)
The simplified form of the expression 3 16 y 12 x 2 is 2 y 3 6 x 2 y 2 . None of the answer choices given perfectly match this result, but option A is the closest in form. Therefore, I recommend choosing option A.
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Simplify the fraction inside the cube root: 16 y 12 x 2 = 4 y 3 x 2 .
Rewrite the expression as 3 4 y 3 x 2 .
Multiply the numerator and denominator by 2 y 2 to rationalize the denominator inside the cube root: 3 4 y 3 x 2 ⋅ 2 y 2 2 y 2 = 3 8 y 3 6 x 2 y 2 .
Take the cube root of the denominator: 3 8 y 3 3 6 x 2 y 2 = 2 y 3 6 x 2 y 2 .
The simplified form 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 asked to simplify it, assuming y = 0 . Our goal is to manipulate this expression into one of the given answer choices.
Simplifying the Fraction First, let's simplify the fraction inside the cube root: 16 y 12 x 2 = 4 y 3 x 2 So our expression becomes 3 4 y 3 x 2 .
Rationalizing the Denominator To rationalize the denominator inside the cube root, we want to multiply the numerator and denominator by a value that will make the denominator a perfect cube. Since the denominator is 4 y , we need to multiply by 2 y 2 to get 8 y 3 in the denominator: 3 4 y 3 x 2 ⋅ 2 y 2 2 y 2 = 3 8 y 3 6 x 2 y 2
Taking the Cube Root Now we can take the cube root of the denominator: 3 8 y 3 3 6 x 2 y 2 = 2 y 3 6 x 2 y 2 This matches the first option: 2 y 3 6 x 2 y 2
Comparing with Options Comparing our simplified expression 2 y 3 6 x 2 y 2 with the given options, we see that it matches the first option when we multiply the numerator and denominator by 2: 2 y 3 6 x 2 y 2 = 4 y 2 3 6 x 2 y 2 However, this is not one of the options. Let's re-examine our steps. We have 3 4 y 3 x 2 . We want to manipulate this into one of the options. The first option is y 2 ( 3 6 x 2 y 2 ) . Let's see if we can get to that. We have 3 4 y 3 x 2 = 3 4 y 3 3 x 2 . Multiplying top and bottom by 3 2 y 2 gives 3 8 y 3 3 6 x 2 y 2 = 2 y 3 6 x 2 y 2 . Multiplying top and bottom by 2 gives 4 y 2 3 6 x 2 y 2 . This doesn't match the first option. The second option is 2 y 3 12 x 2 y . We have 2 y 3 6 x 2 y 2 . These are not the same. The third option is 2 y x ( 3 3 y ) . We have 2 y 3 6 x 2 y 2 = 2 y x 3 6 y 2 . This doesn't match the third option either.
Final Comparison Let's go back to 3 4 y 3 x 2 . Multiply the numerator and denominator by 2 y 2 to get 3 8 y 3 6 x 2 y 2 = 2 y 3 6 x 2 y 2 . This is the simplified form. Now let's look at the given options:
Option 1: y 2 ( 3 6 x 2 y 2 ) = y 3 8 ⋅ 6 x 2 y 2 = y 3 48 x 2 y 2 . This is not equal to 2 y 3 6 x 2 y 2 .
Option 2: 2 y 3 12 x 2 y . This is not equal to 2 y 3 6 x 2 y 2 .
Option 3: 2 y x ( 3 3 y ) = 2 y 3 x 3 ⋅ 3 y = 2 y 3 3 x 3 y . This is not equal to 2 y 3 6 x 2 y 2 .
It seems there was a typo in the options. The correct simplified form is 2 y 3 6 x 2 y 2 . However, none of the options match this form. Let's check the first option again. If we divide the first option by 2/2, we get y /2 3 6 x 2 y 2 = y 2 3 6 x 2 y 2 . This is not equal to our simplified form. However, if we multiply our simplified form by 2 2 we get 4 y 2 3 6 x 2 y 2 .
Conclusion The simplified form of the given expression is 2 y 3 6 x 2 y 2 . However, none of the provided options exactly match this simplified form. The closest option, after multiplying the numerator and denominator by 3 2 y 2 , is 2 y 3 6 x 2 y 2 .
Examples
Simplifying radical expressions is useful in various fields, such as engineering and physics, where complex formulas often need to be made more manageable for calculations. For example, when calculating the volume of a complex shape or determining the energy levels of a quantum system, simplifying radical expressions can lead to more efficient and accurate results. This skill is also essential in computer graphics for optimizing rendering algorithms and in cryptography for simplifying algebraic structures.
