Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ 2^{3} \cdot 2^{5} $$

Step-by-Step Solution

Answer

By applying the exponential property \(a^{m} \cdot a^{n} = a^{m+n}\), we can simplify the given expression: \(2^{3} \cdot 2^{5} = 2^{3+5} = 2^8 = 256\).

Step 1: Identify the exponential properties

In this case, we have two terms with the same base (2) that are being multiplied. The property we will use is: \[a^{m} \cdot a^{n} = a^{m+n}\]

Step 2: Apply the exponential properties

Applying this property to our expression: $$2^{3} \cdot 2^{5} = 2^{3+5}$$

Step 3: Add the exponents

Now, add the exponents (3 and 5): $$2^{3+5} = 2^8$$

Step 4: Evaluate the resulting expression

Finally, evaluate the expression: $$2^8 = 256$$ So, the expression simplifies to 256.

Key Concepts

Exponential Properties

Understanding exponential properties is vital when working with expressions involving exponents. Exponents represent repeated multiplication, allowing us to simplify expressions quickly.
For instance, when you see a term like \(2^3\), it means \(2\times 2 \times 2\), which equals 8. These properties not only make calculations easier but also provide a systematic approach to simplifying broader, complex expressions.
In the context of the exercise, we use the property \(a^m \cdot a^n = a^{m+n}\), which is central to our understanding of exponents. This property tells us that when multiplying powers with the same base, you simply add the exponents.
Recognizing these properties is the first step in evaluating expressions more efficiently.

Multiplying Powers with the Same Base

One of the most powerful applications of exponential properties is multiplying powers that share the same base. As mentioned, when you multiply these powers, you add their exponents.
In the given problem, this translates to the expression \(2^3 \cdot 2^5\). Here, both terms have the base of 2.
By applying the property \(a^m \cdot a^n = a^{m+n}\), we find: \[ 2^3 \cdot 2^5 = 2^{3+5} = 2^8 \]
Adding the exponents (3 and 5) yields 8. This approach greatly simplifies the workload compared to multiplying individual terms. It exemplifies how these concepts help manage more extensive equations efficiently and accurately.

Evaluating Expressions

After simplifying the expression by correctly applying exponential properties, the next step is evaluating the result to find a numerical value.
In our example, we've reduced \(2^3 \cdot 2^5\) to \(2^8\). Now, computing \(2^8\) involves calculating the result of multiplying the base, 2, by itself 7 more times (a total of 8 two's multiplied together).
This results in:

• \(2 \times 2 = 4\)
• \(4 \times 2 = 8\)
• \(8 \times 2 = 16\)
• \(16 \times 2 = 32\)
• \(32 \times 2 = 64\)
• \(64 \times 2 = 128\)
• \(128 \times 2 = 256\)

Ultimately, this means \(2^8 = 256\), which is the final value of the expression. Understanding each step helps reinforce the mathematical concepts and ensures accuracy.