Part C
Rewrite $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}$ as an exponential expression with a base of 2.
Answer (1)
Multiply the fractions: 2 1 \t ⋅ 2 1 \t ⋅ 2 1 = 8 1 .
Recognize that 8 = 2 3 , so 8 1 = 2 3 1 .
Rewrite the reciprocal as a negative exponent: 2 3 1 = 2 − 3 .
The expression 2 1 \t ⋅ 2 1 \t ⋅ 2 1 as an exponential expression with a base of 2 is 2 − 3 .
Explanation
Understanding the Problem We are asked to rewrite the expression 2 1 \t ⋅ 2 1 \t ⋅ 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 , where x is some exponent.
Simplifying the Expression First, let's simplify the given expression by multiplying the fractions: 2 1 \t ⋅ 2 1 \t ⋅ 2 1 = 2 × 2 × 2 1 × 1 × 1 = 8 1
Expressing as a Power of 2 Now, we want to express 8 1 as a power of 2. We know that 8 = 2 × 2 × 2 = 2 3 . Therefore, 8 1 can be written as the reciprocal of 2 3 , which is 2 − 3 . 8 1 = 2 3 1 = 2 − 3
Final Answer Thus, the expression 2 1 \t ⋅ 2 1 \t ⋅ 2 1 can be rewritten as 2 − 3 .
Examples
Exponential expressions are used in various fields, such as calculating compound interest. For example, if you invest 100 a t anann u a l in t eres t r a t eo f 5 t$ years can be expressed as 100 ( 1.05 ) t . This shows how a base (1.05 in this case) is raised to a power to model growth over time.
