Part D
Rewrite $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}$ as an exponential expression with a base of 2.
Answer (1)
Rewrite the expression as ( 2 1 ) 4 .
Express 2 1 as 2 − 1 .
Substitute 2 − 1 into the expression to get ( 2 − 1 ) 4 .
Simplify the expression using the power of a power rule: ( 2 − 1 ) 4 = 2 − 4 . The final answer is 2 − 4 .
Explanation
Understanding the problem We are asked to rewrite the expression 2 1 \t ⋅ 2 1 ⋅ 2 1 ⋅ 2 1 as an exponential expression with a base of 2. This means we want to express the given product in the form 2 x for some exponent x .
Rewriting with exponents First, let's rewrite the given expression using exponents. Since we are multiplying 2 1 by itself four times, we can write this as ( 2 1 ) 4 .
Expressing the base as a power of 2 Now, we want to express 2 1 as a power of 2. Recall that 2 − 1 = 2 1 . So we can substitute 2 − 1 for 2 1 in our expression: ( 2 1 ) 4 = ( 2 − 1 ) 4 .
Simplifying the expression Finally, we use the power of a power rule, which states that ( a m ) n = a m \t ⋅ n . In our case, we have ( 2 − 1 ) 4 = 2 − 1 \t ⋅ 4 = 2 − 4 . Therefore, the expression 2 1 ⋅ 2 1 ⋅ 2 1 ⋅ 2 1 can be written as 2 − 4 .
Examples
Exponential expressions are used in various fields, such as calculating compound interest, modeling population growth, and describing radioactive decay. For instance, if a bacteria population doubles every hour, the population after t hours can be modeled as P ( t ) = P 0 \t ⋅ 2 t , where P 0 is the initial population. Similarly, in finance, compound interest can be calculated using the formula A = P ( 1 + r / n ) n t , where A is the final amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Understanding exponential expressions helps in making predictions and analyzing trends in these real-world scenarios.
