In this blog, we’ll walk through how to evaluate the expression \( 2^5 \). Exponents are a fundamental concept in mathematics, and understanding them starts with examples like this.

### What Does $2^5$ Mean?

The expression \( 2^5 \) means that we multiply the number 2 by itself a total of 5 times. More generally, an expression like \( a^n \) means that the number \( a \), referred to as the base, is multiplied by itself \( n \) times. So for \( 2^5 \), we have

\[\begin{align*} 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2. \end{align*}\]

### Step-by-Step Calculation

Let’s go step by step to calculate \( 2^5 \).

\[\begin{align*} \underline{2 \cdot 2} \cdot 2 \cdot 2 \cdot 2 &= \underline{4 \cdot 2} \cdot 2 \cdot 2 \\ &= \underline{8 \cdot 2} \cdot 2 \\ &= \underline{16 \cdot 2} \\ &= 32. \end{align*}\]

### Final Answer

The value of \( 2^5 \) is \( \boxed{32} \).

### Looking for More?

We are not limited to taking the exponent of whole numbers. For example, we could calculate exponents with a base of a negative number, such as $(-3)^3$:

\[\begin{align*} \underline{(-3) \cdot (-3)} \cdot (-3) &= \underline{9 \cdot (-3)} \\ &= -27. \end{align*}\]

In the same way, we could calculate $\left( \dfrac{1}{2} \right)^{3}$:

\[\begin{align*} \underline{\left( \dfrac{1}{2} \right) \cdot \left( \dfrac{1}{2} \right)} \cdot \dfrac{1}{2} &= \underline{\left( \dfrac{1}{4} \right) \cdot \left( \dfrac{1}{2} \right)} \\ &= \dfrac{1}{8}. \end{align*}\]

Now, we might be tempted to ask what happens if we take our numbers to a negative exponent, or even a fractional exponent. To learn and practice more with exponents and other advanced math topics, check out the “I Aced Calculus” app. It offers tons of practice problems and flashcards to help you master these concepts quickly!