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)
The simplified form of the expression 4 2 x 3 is 2 x 4 24 x 3 , which matches option B. To achieve this, we multiplied the numerator and denominator by a factor to eliminate the fraction in the fourth root. The process involved simplifying the fourth root of both the numerator and the denominator separately.
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Multiply the numerator and denominator of the fraction inside the fourth root 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 .
Simplify the expression: 4 16 x 4 24 x 3 = 4 16 x 4 4 24 x 3 = 2 x 4 24 x 3 .
The simplified form of the given expression is 2 x 4 24 x 3 .
Explanation
Understanding the Problem We are given the expression 4 2 x 3 and the condition 0"> x > 0 . Our goal is to simplify this expression and choose the correct simplified form from the options provided.
Eliminating the Fraction Inside the Root To simplify the expression, we want to eliminate the fraction inside the fourth root. We can do this by multiplying the numerator and the denominator by a factor that will make the denominator a perfect fourth power. Since we have 2 x in the denominator, we need to multiply by 8 x 3 to get 16 x 4 = ( 2 x ) 4 in the denominator.
Multiplying by the Appropriate Factor Multiplying the numerator and denominator by 8 x 3 , we get: 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: 4 16 x 4 24 x 3 = 4 16 x 4 4 24 x 3 = 2 x 4 24 x 3
Choosing the Correct Option Comparing the simplified expression 2 x 4 24 x 3 with the given options, we see that it matches option 2 x 4 24 x 3 .
Final Answer Therefore, the simplified form of the given expression is 2 x 4 24 x 3 .
Examples
Imagine you are working in a lab and need to prepare a solution with a specific concentration. The formula for the concentration involves a fourth root expression similar to the one we simplified. By simplifying the expression, you can more easily calculate the required amounts of chemicals to achieve the desired concentration. This type of simplification is useful in various scientific and engineering applications where complex formulas need to be evaluated efficiently.
