What is the simplified form of the following expression? Assume [tex]x\ \textgreater \ 0[/tex]. [tex]\sqrt[4]{\frac{3}{2 x}}[/tex]
A. [tex]\frac{\sqrt[4]{6 x}}{2 x}[/tex]
B. [tex]\frac{\sqrt[4]{24 x^3}}{2 x}[/tex]
C. [tex]\frac{\sqrt[4]{24 x^3}}{16 x^4}[/tex]
D. [tex]\sqrt[4]{12 x^2}[/tex]
Answer (2)
Rationalize the denominator of the expression inside the fourth root by multiplying the numerator and denominator by 8 x 3 .
Simplify the expression: 4 2 x 3 = 4 16 x 4 24 x 3 .
Take the fourth root of the denominator: 4 16 x 4 4 24 x 3 = 2 x 4 24 x 3 .
The simplified form of the expression is 2 x 4 24 x 3 .
Explanation
Understanding the Problem We are given the expression 4 2 x 3 and we want to simplify it, assuming that 0"> x > 0 . Our goal is to manipulate the expression to match one of the given options. The key idea here is to rationalize the denominator inside the fourth root.
Rationalizing the Denominator To rationalize the denominator, we want to make the denominator a perfect fourth power. We have 2 x in the denominator. To make it a perfect fourth power, we need to multiply it by 2 3 x 3 = 8 x 3 . So, we multiply both the numerator and the denominator by 8 x 3 :
Multiplying by 8 x 3 4 2 x 3 = 4 2 x ⋅ 8 x 3 3 ⋅ 8 x 3 = 4 16 x 4 24 x 3
Simplifying the Expression Now we can simplify the expression by taking the fourth root of the denominator:
Taking the Fourth Root 4 16 x 4 24 x 3 = 4 16 x 4 4 24 x 3 = 2 x 4 24 x 3
Final Answer Comparing our simplified expression 2 x 4 24 x 3 with the given options, we see that it matches the second option.
Examples
Imagine you are a pharmacist and need to dilute a certain medicine to a specific concentration. Simplifying radical expressions like this helps you calculate the exact amount of diluent needed to achieve the desired concentration. This ensures the medicine is safe and effective for the patient. Similarly, in engineering, simplifying such expressions can help in calculating stress or strain in materials, ensuring structural integrity.
The simplified form of 4 2 x 3 is 2 x 4 24 x 3 , which corresponds to option B. The process involves rationalizing the denominator to make it a perfect fourth power and then simplifying. Thus, the final answer is option B.
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