Part B
Rewrite $\frac{1}{2} \cdot \frac{1}{2}$ as an exponential expression with a base of 2.
Answer (1)
Simplify the expression: 2 1 \t ⋅ 2 1 = 4 1 .
Rewrite 4 1 as a power of 2: 4 1 = 2 2 1 = 2 − 2 .
Therefore, 2 1 ⋅ 2 1 = 2 − 2 .
The exponential expression is 2 − 2 .
Explanation
Understanding the Problem We are asked to rewrite the expression 2 1 ⋅ 2 1 as an exponential expression with a base of 2. This means we want to find an exponent x such that 2 x = 2 1 ⋅ 2 1 .
Simplifying the Expression First, let's simplify the expression 2 1 ⋅ 2 1 . When we multiply fractions, we multiply the numerators and the denominators: 2 1 ⋅ 2 1 = 2 ⋅ 2 1 ⋅ 1 = 4 1
Rewriting as a Power of 2 Now, we want to express 4 1 as a power of 2. We are looking for x such that 2 x = 4 1 . We know that 2 2 = 4 , so 4 1 = 2 2 1 . Using the property that a n 1 = a − n , we can rewrite 2 2 1 as 2 − 2 . Therefore, 2 x = 2 − 2 , which means x = − 2 .
Final Answer Thus, we can rewrite 2 1 ⋅ 2 1 as 2 − 2 .
Examples
Exponential expressions are used in various real-world applications, such as calculating compound interest, modeling population growth, and describing radioactive decay. For example, if you invest money in an account that pays compound interest, the amount of money you have after a certain period can be calculated using an exponential expression. Similarly, the decay of a radioactive substance can be modeled using an exponential expression, where the base represents the fraction of the substance that remains after each time period.
