What is the simplified form of the following expression? Assume [tex]$x \neq 0$[/tex].
[tex]$\sqrt[5]{\frac{10 x}{3 x^3}}$[/tex]
Answer (2)
To simplify 5 3 x 3 10 x , start by simplifying the fraction to 3 x 2 10 . Then rationalizing the denominator leads to the final form 3 x 5 810 x 3 .
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Simplify the fraction inside the radical: 3 x 3 10 x = 3 x 2 10 .
Multiply the numerator and denominator inside the radical by 3 4 x 3 to rationalize the denominator: 5 3 x 2 10 = 5 3 5 x 5 10 × 3 4 x 3 .
Simplify the expression: 3 x 5 10 × 3 4 x 3 = 3 x 5 810 x 3 .
The simplified form of the expression is 3 x 5 810 x 3 .
Explanation
Understanding the Problem We are given the expression 5 3 x 3 10 x and the condition x = 0 . Our goal is to simplify this expression.
Simplifying the Fraction First, we simplify the fraction inside the radical by canceling the common factor x from the numerator and the denominator: 3 x 3 10 x = 3 x 2 10 So the expression becomes 5 3 x 2 10 .
Rationalizing the Denominator To rationalize the denominator inside the radical, we want to eliminate the x 2 term in the denominator. Since we are taking the fifth root, we need to have x 5 in the denominator. To achieve this, we multiply both the numerator and the denominator by 3 4 x 3 inside the radical: 5 3 x 2 10 = 5 3 x 2 × 3 4 x 3 10 × 3 4 x 3 = 5 3 5 x 5 10 × 3 4 x 3
Simplifying the Fifth Root Now we can rewrite the expression as: 5 3 5 x 5 10 × 3 4 x 3 = 5 3 5 x 5 5 10 × 3 4 x 3 = 3 x 5 10 × 81 x 3 = 3 x 5 810 x 3
Final Answer Therefore, the simplified form of the given expression is 3 x 5 810 x 3 .
Examples
Imagine you are working in a lab and need to prepare a solution with a specific concentration. Simplifying radical expressions like this can help you determine the exact amount of a substance to add to a solvent. For instance, if you have a stock solution with a concentration represented by 5 3 x 3 10 x , simplifying it to 3 x 5 810 x 3 allows for easier calculation and measurement of the required volume. This ensures accuracy and efficiency in your experimental procedures.
