Rewrite $42^{\frac{5}{4}}$ as a radical.
A. $(\sqrt[5]{42})^4$
B. $\sqrt[5]{42^4}$
C. $\sqrt[1.25]{42}$
D. $(\sqrt[4]{42})^5$
Rewrite $\sqrt{\left(\frac{3}{4}\right)^9}$ as a power (exponential form).
A. $\left(\frac{3}{4}\right)^{\frac{2}{9}}$
B. $\left(\frac{3}{4}\right)^{-\frac{9}{2}}$
C. $\left(\frac{3}{4}\right)^{\frac{9}{2}}$
D. $\left(\frac{4}{3}\right)^{-\frac{2}{9}}$
Answer (1)
Rewrite 4 2 4 5 as a radical: 4 2 4 5 = ( 4 42 ) 5 .
Rewrite ( 4 3 ) 9 as a power: ( 4 3 ) 9 = ( 4 3 ) 2 9 .
The answer to Question 3 is ( 4 42 ) 5 .
The answer to Question 4 is ( 4 3 ) 2 9 .
Explanation
Problem Analysis We will address each question separately, providing a step-by-step conversion to the required form.
Question 3 Solution For Question 3, we need to rewrite 4 2 4 5 as a radical. Recall the property a n m = ( n a ) m = n a m . Applying this property, we have 4 2 4 5 = ( 4 42 ) 5 .
Question 4 Solution For Question 4, we need to rewrite ( 4 3 ) 9 as a power. Recall that a = a 2 1 . Therefore, ( 4 3 ) 9 = ( 4 3 ) 2 9 .
Final Answer Therefore, the answer to Question 3 is ( 4 42 ) 5 , and the answer to Question 4 is ( 4 3 ) 2 9 .
Examples
Understanding fractional exponents and radicals is crucial in various fields, such as physics and engineering, where they are used to model phenomena like wave propagation and heat transfer. For instance, the period of a pendulum can be expressed using a fractional exponent, and understanding how to manipulate these expressions allows engineers to design more efficient systems. Similarly, in finance, compound interest calculations often involve fractional exponents, enabling analysts to predict investment growth accurately. By mastering these concepts, students gain a valuable toolset for solving real-world problems across diverse disciplines.
