Which expression is the same as [tex]$x^{\frac{7}{3}}$[/tex]?
A. [tex]$\sqrt[7]{x^3}$[/tex]
B. [tex]$\sqrt[3]{x^7}$[/tex]
C. [tex]$\frac{1}{7} x^3$[/tex]
D. [tex]$\frac{1}{3} x^7$[/tex]
Answer (2)
Rewrite the fractional exponent as a radical: x n m = n x m .
Apply the rule to the given expression: x 3 7 = 3 x 7 .
Compare the result with the given options.
The equivalent expression is 3 x 7 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to x 3 7 . Let's recall the relationship between fractional exponents and radicals.
Recalling the Rule Recall that x n m can be written as n x m , where m is the power and n is the index of the radical. In our case, we have x 3 7 , so m = 7 and n = 3 .
Applying the Rule to the Problem Applying this rule, we can rewrite x 3 7 as 3 x 7 . This means we are taking the cube root of x raised to the power of 7.
Comparing with Options Now, let's compare our result, 3 x 7 , with the given options:
7 x 3 - This is x 7 3 , which is not the same.
3 x 7 - This matches our result.
7 1 x 3 - This is a linear transformation, not an exponent.
3 1 x 7 - This is also a linear transformation, not an exponent.
Therefore, the correct expression is 3 x 7 .
Final Answer The expression equivalent to x 3 7 is 3 x 7 .
Examples
Understanding fractional exponents is crucial in various fields, such as physics and engineering. For instance, when calculating the period of a pendulum, the formula involves a square root, which can be expressed as a fractional exponent. Similarly, in fluid dynamics, the velocity of a fluid flowing through a pipe might involve fractional exponents when dealing with turbulent flow. Knowing how to manipulate and simplify these expressions allows engineers and scientists to make accurate predictions and design efficient systems. For example, the expression x 3 7 could represent a scaled relationship between two physical quantities, such as volume and pressure in a thermodynamic system, where understanding this relationship helps in optimizing system performance.
The expression equivalent to x 3 7 is 3 x 7 , which corresponds to option B. This conversion uses the relationship between fractional exponents and radicals. Therefore, the correct answer is 3 x 7 .
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