Which expression is equivalent to $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ ? Assume $x \neq 0$ and $y>0$.
A. $\frac{\sqrt[4]{3}}{2 x^2 y}$
B. $\frac{x(\sqrt[4]{3})}{4 y^2}$
C. $\frac{\sqrt[4]{3}}{4 x y^2}$
Answer (2)
Simplify the fraction inside the radical: 128 x 4 y 5 24 x 6 y = 16 y 4 3 x 2 .
Substitute the simplified fraction back into the radical: 4 16 y 4 3 x 2 .
Simplify the radical expression: 4 16 y 4 4 3 x 2 = 2 y 4 3 x .
None of the given options match the simplified expression, indicating a possible typo in the original question or answer choices.
Explanation
Simplifying the Fraction We are asked to simplify the expression 4 128 x 4 y 5 24 x 6 y assuming that x = 0 and 0"> y > 0 . Let's start by simplifying the fraction inside the radical.
Simplifying the Numerical Fraction First, we simplify the numerical fraction 128 24 . Both 24 and 128 are divisible by 8, so we have 128 24 = 128 ÷ 8 24 ÷ 8 = 16 3 .
Simplifying the x terms Next, we simplify the expression with x . We have x 4 x 6 = x 6 − 4 = x 2 .
Simplifying the y terms Now, we simplify the expression with y . We have y 5 y = y 1 − 5 = y − 4 = y 4 1 .
Combining the Simplified Terms Combining these results, we have 128 x 4 y 5 24 x 6 y = 16 y 4 3 x 2 .
Substituting Back into the Radical Now we substitute this back into the original expression: 4 128 x 4 y 5 24 x 6 y = 4 16 y 4 3 x 2 .
Separating the Radical We can rewrite the fourth root as 4 16 y 4 3 x 2 = 4 16 y 4 4 3 x 2 .
Simplifying the Denominator We know that 4 16 = 2 and 4 y 4 = y since 0"> y > 0 . Also, 4 x 2 = ( x 2 ) 4 1 = x 4 2 = x 2 1 = x . Thus, 4 16 y 4 = 2 y .
Combining the Results Therefore, we have 4 16 y 4 4 3 x 2 = 2 y 4 3 x 2 . We can also write this as 2 y 4 3 ⋅ 4 x 2 = 2 y 4 3 x . However, none of the answer choices match this form.
Re-examining the Expression Let's re-examine the expression 2 y 4 3 x 2 . We can rewrite x 2 as ( x ) 4 , so 4 3 x 2 = 4 3 ( x ) 4 . However, this does not lead to any of the given options.
Analyzing the Options Let's analyze the given options. We have: Option 1: 2 x 2 y 4 3 Option 2: 4 y 2 x ( 4 3 ) Option 3: 4 x y 2 4 3 Comparing these to our simplified expression 2 y 4 3 x 2 = 2 y 4 3 x , we see that none of the options are equivalent. However, let's reconsider our simplification. We have 4 16 y 4 3 x 2 = 2 y 4 3 x . If the problem meant to have x 6 in the denominator instead of x 4 then we would have 4 128 24 x 6 x 6 y 5 y = 4 16 y 4 3 = 2 y 4 3 . But this is not the case.
Final Answer We have 4 16 y 4 3 x 2 = 2 y 4 3 x . None of the options match this. However, if we look at option 2, 4 y 2 x ( 4 3 ) , we can see that it is the closest if we assume that there was a typo in the original question. However, we must stick to the original question.
Conclusion Since none of the options are equivalent to the simplified expression, there must be a typo in the original question or the answer choices. However, based on the given information, none of the provided options are correct.
Examples
Simplifying radical expressions is useful in various fields such as physics and engineering when dealing with complex equations. For example, when calculating the energy levels of a quantum mechanical system, you might encounter expressions involving radicals that need to be simplified to obtain a more manageable form. Similarly, in electrical engineering, simplifying radical expressions can help in analyzing circuits and determining impedance values. These simplifications make calculations easier and provide a clearer understanding of the underlying relationships.
The expression 4 128 x 4 y 5 24 x 6 y simplifies to 2 y 4 3 x . None of the presented options match the simplified expression, indicating a possible issue with the choices. Therefore, none of the options provided are correct.
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