Problem 13 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$2^{2} \cdot 2^{3}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(2^{2} \cdot 2^{3}\) is 32.

Step 1: Identify the Bases and Exponents

In the given expression \(2^{2} \cdot 2^{3}\), both 2's are bases and 2 and 3 are their respective exponents.

Step 2: Apply the Law of Exponents

According to the law of exponents, when multiplying terms with the same base, we keep the base and add the exponents. So, \(2^{2} \cdot 2^{3}\) = \(2^{2+3}\).

Step 3: Add the Exponents

Adding the exponents gives us \(2^{5}\).

Step 4: Evaluate the Expression

Now, calculate \(2^{5}\). The base number 2 is multiplied by itself 5 times in total which equals 32.

Key Concepts

Law of Exponents

Understanding the law of exponents is crucial when dealing with exponential expressions. It's a set of rules that describe how to handle these expressions when they are multiplied, divided, or raised to a power. For instance, when multiplying two expressions with the same base, as in the exercise, the rule is simple: keep the base and add the exponents together. To put it into formula terms, we write it as \( a^m \cdot a^n = a^{m+n} \).

Applying this law makes calculations much simpler and more efficient. Instead of multiplying the base number multiple times, one can directly combine the exponents and compute the final result. This rule is foundational, and there are additional laws for situations where the base or exponent is different, or when dividing and raising powers to powers.

Evaluating Exponentials

When evaluating exponentials, one essentially computes the value of a number raised to a certain power. The goal is to understand how large or small a number will become when repeated multiplication (or division, for negative exponents) is applied. Taking our exercise as an example, evaluating \(2^{5}\) means calculating the value of 2 multiplied by itself a total of five times. This results in 2 × 2 × 2 × 2 × 2, which equals 32.

To simplify the process and avoid manual calculation, one can use the laws of exponents initially to reduce complex exponential expressions. Then, for the actual evaluating step, depending on the size of the exponent, either manual calculation or a calculator can be used. Understanding how to evaluate exponentials is not only a mathematical skill but essential for real-world applications, such as computing compound interest or understanding exponential growth.

Base and Exponent

In exponential expressions, the base and the exponent are two fundamental components. The base is the number that is being multiplied by itself, and the exponent tells us how many times the multiplication occurs. In our problem, 2 is the base and 5 is the exponent in \(2^{5}\).

The base is the repeated factor in the multiplication process—what we are calculating power for. The exponent, on the other hand, is the small raised number that indicates the number of times the base is used as a factor in the multiplication. It's essential to distinguish between the two when applying the laws of exponents because each plays a very different role in the expression's evaluation and simplification.