Which of the following is equivalent to $(5)^{\frac{7}{3}}$?
A. $5^{-4}$
B. $5^4$
C. $\sqrt[7]{5^3}$
D. $\sqrt[3]{5^7}$
Answer (1)
Rewrite the expression ( 5 ) 3 7 using radicals.
Recall that a n m = n a m .
Apply this rule to rewrite ( 5 ) 3 7 as 3 5 7 .
The equivalent expression is 3 5 7 .
Explanation
Understanding the Problem We are given the expression ( 5 ) 3 7 and asked to find an equivalent expression from the given choices.
Recalling the Rule We need to rewrite the given expression using radicals. Recall that a n m = n a m .
Applying the Rule Applying this rule to rewrite ( 5 ) 3 7 as 3 5 7 .
Finding the Equivalent Expression Comparing the result with the given choices, we see that 3 5 7 is one of the options.
Final Answer Therefore, the expression equivalent to ( 5 ) 3 7 is 3 5 7 .
Examples
Understanding fractional exponents is crucial in various fields like physics and engineering. For example, when calculating the period of a pendulum, you might encounter expressions with fractional exponents. Similarly, in electrical engineering, analyzing circuit behavior often involves dealing with impedance calculations that include fractional powers. Knowing how to convert between fractional exponents and radicals allows for easier manipulation and simplification of these expressions, leading to more accurate and efficient problem-solving.
