$\begin{array}{l}\left(\frac{1}{125}\right)^{\frac{2}{3}}=\square \\ 64^{-\frac{2}{3}}=\square\end{array}$
Answer (2)
Rewrite the first expression: ( 125 1 ) 3 2 = ( 5 − 3 ) 3 2 = 5 − 2 .
Simplify the first expression: 5 − 2 = 25 1 = 0.04 .
Rewrite the second expression: 6 4 − 3 2 = ( 4 3 ) − 3 2 = 4 − 2 .
Simplify the second expression: 4 − 2 = 16 1 = 0.0625 . The final answers are 25 1 and 16 1 .
Explanation
Problem Analysis We are asked to evaluate two expressions involving fractional exponents. Let's tackle them one at a time.
Evaluating the First Expression First, we have ( 125 1 ) 3 2 . We can rewrite 125 1 as 12 5 − 1 . Also, we know that 125 = 5 3 . Therefore, we have
( 125 1 ) 3 2 = ( 12 5 − 1 ) 3 2 = ( 5 3 ) − 1 × 3 2 = 5 3 ×− 1 × 3 2 = 5 − 2 = 5 2 1 = 25 1 = 0.04.
Evaluating the Second Expression Next, we have 6 4 − 3 2 . We know that 64 = 4 3 . Therefore, we have
6 4 − 3 2 = ( 4 3 ) − 3 2 = 4 3 ×− 3 2 = 4 − 2 = 4 2 1 = 16 1 = 0.0625.
Final Answer Therefore, we have
( 125 1 ) 3 2 = 25 1 and 6 4 − 3 2 = 16 1 .
Examples
Fractional exponents are used in various fields, such as calculating growth rates or decay rates. For example, if a population grows by a factor of 8 over 27 years, the annual growth rate can be found using fractional exponents. Similarly, in finance, compound interest calculations often involve fractional exponents to determine the return on investment over different time periods. Understanding fractional exponents helps in modeling and predicting changes in these real-world scenarios.
The solutions for the expressions are ( 125 1 ) 3 2 = 25 1 and 6 4 − 3 2 = 16 1 . Each expression involves rewriting bases with exponents and simplifying accordingly. Fractional exponents allow us to express roots and powers in a concise way.
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