Evaluate $0.16^{\frac{5}{2}}$ and $\left(\frac{2}{3}\right)^{-3}$.
Answer (1)
Evaluate 0.1 6 2 5 by rewriting 0.16 as 25 4 and simplifying to ( 5 2 ) 5 , which equals 0.01024 .
Evaluate ( 3 2 ) − 3 by taking the reciprocal and cubing, resulting in ( 2 3 ) 3 = 8 27 .
The first expression evaluates to 0.01024 .
The second expression evaluates to 8 27 .
Explanation
Problem Analysis We are asked to evaluate two expressions: 0.1 6 2 5 and ( 3 2 ) − 3 . Let's tackle them one at a time.
Evaluating the first expression First, let's evaluate 0.1 6 2 5 . We can rewrite 0.16 as a fraction: 0.16 = 100 16 = 25 4 . Now we have ( 25 4 ) 2 5 . This can be expressed as ( 25 4 ) 5 . The square root of 25 4 is 5 2 . So we have ( 5 2 ) 5 = 5 5 2 5 = 3125 32 . Converting this fraction to a decimal, we get 0.01024 .
Evaluating the second expression Next, let's evaluate ( 3 2 ) − 3 . A negative exponent means we take the reciprocal of the base and raise it to the positive exponent. So, ( 3 2 ) − 3 = ( 2 3 ) 3 = 2 3 3 3 = 8 27 . Converting this fraction to a decimal, we get 3.375 .
Final Answer Therefore, 0.1 6 2 5 = 0.01024 and ( 3 2 ) − 3 = 8 27 .
Examples
Understanding exponents and fractions is crucial in many real-world scenarios. For instance, calculating compound interest involves raising a principal amount to a power that represents the number of compounding periods. Similarly, in physics, inverse relationships often involve negative exponents, such as the relationship between pressure and volume in Boyle's Law. Mastering these concepts allows for accurate modeling and prediction in various scientific and financial applications.
