Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ 3^{4} $$

Step-by-Step Solution

Answer

The expression \(3^4\) represents the base (3) multiplied by itself for a number of times equal to the exponent (4). Therefore, it is calculated as \(3 \times 3 \times 3 \times 3 = 81\).

Step 1: Understand the exponentiation notation

In this expression, \(3^4\), the base number is 3 and the exponent is 4. This notation means that we need to multiply the base (3) by itself for a number of times equal to the exponent (4).

Step 2: Multiply the base by itself

Since the exponent is 4, we need to multiply the base (3) by itself 4 times. It looks like this: \[3 \times 3 \times 3 \times 3\]

Step 3: Perform the multiplications

When multiplying, it's best to start with the first pair of numbers and work your way through the list. In our case, we should multiply 3 by 3 and then multiply the result by 3, and so on: \[3 \times 3 = 9\] \[9 \times 3 = 27\] \[27 \times 3 = 81\]

Step 4: Write down the final result

The final result of the expression \(3^4\) is 81.

Key Concepts

Base Number

The base number is a fundamental component in the concept of exponentiation. In our example, the base number is 3. The base represents the number that is being multiplied. It serves as the foundational figure that is repeated during the multiplication process.

• The base number is always placed at the bottom of the exponential notation, as in \(3^4\).
• The larger and repeated this base is, the higher the resulting value from the calculation.

Understanding the base number is crucial because without it, the operation of exponentiation cannot be completed. Consider it as the cornerstone upon which the structure of multiplication is built.

Exponent

The exponent is the small number or symbol located at the upper right of the base number in the expression. In \(3^4\), the 4 is the exponent. It indicates how many times the base number should be multiplied by itself.

• Exponents are a compact way to show repeated multiplication, saving space and simplifying calculations.
• The exponent changes the value exponentially as it increases; hence, even a small rise in the exponent results in a much larger number.

Think of the exponent as the instruction manual telling you how many times to use the base in a multiplication sequence. In our example, the exponent directs that the base number 3 is to be used four times.

Multiplication

Multiplication is the mathematical process used to calculate the expression once the base and exponent have been identified. In the expression \(3^4\), you are performing multiplication multiple times.

• The process begins by multiplying the base number by itself.
• For \(3^4\), you have to multiply 3 by 3, then take the result and multiply by 3 again, repeating this process until you multiply the base a total of four times.

Let's break it down for our expression:First, multiply: \[3 \times 3 = 9\] Next, continue multiplying: \[9 \times 3 = 27\] Finally, one more multiplication: \[27 \times 3 = 81\] This step-by-step multiplication reveals how the expression is evaluated and why understanding multiplication is critical for calculating powers.

Power

The power is the result of raising a base number to an exponent. When we calculate \(3^4\), the power is the answer we get from the computation, which is 81 in this scenario.

• The power reflects the product of the base number repeated according to the exponent's instruction.
• It provides the end value of an exponential operation.

Recognizing the power helps in understanding the end result of any exponentiation process. It's important to realize that even a small base or exponent can lead to a large power, demonstrating the significant impact of exponentiation.