Which of the following represents $\sqrt[4]{x^3}$ in exponential form?
A. $x^{\frac{3}{4}}$
B. $x^{\frac{4}{3}}$
C. $4 x^3$
D. $3 x^4$
Answer (2)
The exponential form of 4 x 3 is x 4 3 . Thus, the correct answer to the multiple-choice question is option A: x 4 3 .
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Recognize that the problem requires converting a radical expression to exponential form.
Recall the property n x m = x n m .
Apply the property to the given expression 4 x 3 to get x 4 3 .
The exponential form is x 4 3 .
Explanation
Understanding the problem We are asked to express the radical expression 4 x 3 in exponential form.
Recalling the property Recall that a radical expression n x m can be written in exponential form as x n m , where n is the index of the radical and m is the exponent of the radicand.
Applying the property In the given expression, 4 x 3 , the index n is 4 and the exponent m is 3. Therefore, we can rewrite the expression as x 4 3 .
Final Answer The exponential form of 4 x 3 is x 4 3 .
Examples
Understanding how to convert between radical and exponential forms is useful in various scientific and engineering applications. For example, when dealing with physical quantities that scale with fractional exponents, such as in fluid dynamics or heat transfer, converting to exponential form simplifies calculations and analysis. Consider the equation for the period of a simple pendulum, T = 2 π g L , where T is the period, L is the length, and g is the acceleration due to gravity. Rewriting this as T = 2 π ( g L ) 2 1 allows for easier manipulation and comparison with other equations involving exponents.
