$(32^{-3})^{5^{-1}}$
Answer (1)
Rewrite 32 as 2 5 and 5 − 1 as 5 1 .
Apply the power of a power rule: ( a m ) n = a mn .
Simplify the expression to 2 − 3 .
Rewrite 2 − 3 as 2 3 1 and calculate the final value: 8 1 .
Explanation
Understanding the Problem We are given the expression ( 3 2 − 3 ) 5 − 1 and our goal is to simplify it.
Rewriting the Expression First, let's rewrite 32 as 2 5 and 5 − 1 as 5 1 . Substituting these values into the expression, we get ( ( 2 5 ) − 3 ) 5 1 .
Applying the Power of a Power Rule Now, we use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule to the inner part of the expression, we have ( 2 5 ) − 3 = 2 5 ⋅ ( − 3 ) = 2 − 15 . So, the expression becomes ( 2 − 15 ) 5 1 .
Applying the Power of a Power Rule Again Applying the power of a power rule again, we get 2 − 15 ⋅ 5 1 = 2 − 3 .
Rewriting with a Positive Exponent Next, we rewrite 2 − 3 as 2 3 1 .
Calculating the Final Value Finally, we calculate 2 3 = 2 ⋅ 2 ⋅ 2 = 8 , so the expression simplifies to 8 1 .
Final Answer Therefore, the simplified expression is 8 1 .
Examples
Understanding how to simplify exponents is useful in many areas, such as calculating compound interest or dealing with exponential growth or decay in science. For example, if you invest money with an annual interest rate, the formula for the future value involves exponents. Simplifying these expressions helps you understand how your investment grows over time. Similarly, in physics, radioactive decay is modeled using exponential functions, and simplifying exponents helps in determining the remaining amount of a substance after a certain period.
