Which is another way to write $x^{\frac{4}{3}}$?
A. $\sqrt[4]{x^3}$
B. $\frac{1}{3} x^4$
C. $\frac{1}{4} x^3$
D. $\sqrt[3]{x^4}$
Answer (2)
Rewrite the fractional exponent as a radical: x n m = n x m .
Identify m and n in the given expression x 3 4 , so m = 4 and n = 3 .
Substitute m and n into the radical expression: x 3 4 = 3 x 4 .
The equivalent expression is 3 x 4 .
Explanation
Understanding the Problem We are given the expression x 3 4 and asked to find an equivalent expression from the given choices.
Fractional Exponents and Radicals The expression involves a fractional exponent, which can be rewritten using radicals. The general rule is x n m = n x m .
Applying the Rule In this case, we have x 3 4 . Comparing this with the general form x n m , we can identify m = 4 and n = 3 . Therefore, we can rewrite the expression as 3 x 4 .
Finding the Correct Option Now, we compare our result, 3 x 4 , with the given options:
4 x 3 3 1 x 4 4 1 x 3 3 x 4
The correct option is 3 x 4 .
Final Answer The equivalent expression to x 3 4 is 3 x 4 .
Examples
Fractional exponents are used in various fields, such as physics and engineering, to describe relationships between quantities. For example, the period of a simple pendulum is proportional to the square root of its length, which can be expressed as a fractional exponent. Understanding how to convert between fractional exponents and radicals is essential for solving problems involving these relationships. Imagine you are designing a suspension bridge and need to calculate the tension in the cables. The formula for tension might involve fractional exponents, and being able to rewrite these exponents as radicals can simplify the calculations and make the design process more efficient. This skill is also useful in fields like finance, where compound interest calculations often involve fractional exponents.
To rewrite x 3 4 , we can express it as 3 x 4 using the rule that links fractional exponents to radicals. The equivalent expression we found is among the given options. Therefore, the correct answer is D. 3 x 4 .
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