Problem 20 - College Algebra Essentials

Question

Evaluate each exponential expression. $$2^{-3} \cdot 2$$

Step-by-Step Solution

Answer

\(\frac{1}{4}\)

Step 1: Understanding Negative Exponents

By definition, \(a^{-n} = \frac{1}{a^n}\). So in this case, \(2^{-3} = \frac{1}{2^3}\). This changes \(2^{-3} \cdot 2\) into \(\frac{1}{2^3} \cdot 2\).

Step 2: Simplify the Expression

Now we simply have to handle a bit of basic arithmetic. \(2^3 = 8\), so \(\frac{1}{2^3} = \frac{1}{8}\). Then we multiply \(\frac{1}{8}\) by \(2\), which equals \(\frac{2}{8} = \frac{1}{4}\).

Step 3: Check Your Answer

We've simplified \(2^{-3} \cdot 2\) down to \(\frac{1}{4}\). To check our answer, we can remember that a negative exponent essentially means divide, so our expression should be equivalent to \(\frac{2}{2^3}\), which also equals \(\frac{1}{4}\). Our answer is correct.