$(32^{-1})^{5^{-1}}
Answer (2)
The expression ( 3 2 − 1 ) 5 − 1 simplifies to 2 1 . This is achieved by rewriting the expression using fractional exponents and applying the properties of exponents step-by-step. The final result is obtained after simplifying the expression to express 32 as a power of 2 and applying the power rules.
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Rewrite the expression using fractional exponents: ( 3 2 − 1 ) 5 − 1 = ( 3 2 − 1 ) 5 1 = 3 2 − 5 1 .
Express 32 as a power of 2: 3 2 − 5 1 = ( 2 5 ) − 5 1 .
Apply the power of a power rule: ( 2 5 ) − 5 1 = 2 5 × ( − 5 1 ) = 2 − 1 .
Simplify the expression: 2 − 1 = 2 1 . The final answer is 2 1 .
Explanation
Understanding the Problem We are given the expression ( 3 2 − 1 ) 5 − 1 and our goal is to simplify it.
Rewriting with Fractional Exponents First, let's rewrite the expression using fractional exponents. Recall that 5 − 1 = 5 1 , so we have ( 3 2 − 1 ) 5 − 1 = ( 3 2 − 1 ) 5 1
Expressing 32 as a Power of 2 Next, we can rewrite this as 3 2 − 5 1 . Notice that 32 = 2 5 , so we can substitute this into our expression: 3 2 − 5 1 = ( 2 5 ) − 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 m × n . Applying this rule, we get ( 2 5 ) − 5 1 = 2 5 × ( − 5 1 ) = 2 − 1
Simplifying the Expression Finally, we rewrite 2 − 1 as a fraction. Recall that a − n = a n 1 , so 2 − 1 = 2 1 1 = 2 1 Therefore, the simplified expression is 2 1 .
Final Answer The simplified form of the expression ( 3 2 − 1 ) 5 − 1 is 2 1 .
Examples
Understanding exponents and fractional powers is crucial in many scientific fields, such as physics and engineering. For example, when calculating the decay rate of radioactive materials or analyzing the growth of populations, exponential functions and their properties are essential. Simplifying expressions like the one in this problem helps in making these calculations more manageable and understandable. Imagine you are calculating the half-life of a substance; you might encounter similar expressions involving fractional exponents.
