Problem 4 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(4^{3}\)

Step-by-Step Solution

Answer

The value of the expression \(4^3\) is 64.

Step 1: Understand the Problem

We need to find the value of the expression \(4^3\). This expression is an example of exponentiation, which involves a base number raised to the power of an exponent.

Step 2: Identify the Base and Exponent

In the expression \(4^3\), the base is 4 and the exponent is 3. The base, 4, tells us which number is being multiplied, while the exponent, 3, tells us how many times the base is multiplied by itself.

Step 3: Apply Exponentiation

To find the value of \(4^3\), we multiply the base number 4 by itself a total of three times: \(4 \times 4 \times 4\).

Step 4: Perform the Calculations

Calculate \(4 \times 4 = 16\). Then, multiply this result by 4: \(16 \times 4 = 64\).

Step 5: Write the Final Answer

Thus, the value of the expression \(4^3\) is 64.

Key Concepts

Base and Exponent

In mathematical expressions involving powers, two key components are the base and the exponent. Understanding these is essential to perform calculations. The **base** is the number that is being multiplied. It is the principal value in the expression. In our example, the base is 4, which serves as the foundation for the multiplication performed. The **exponent**, on the other hand, is the small number written above and to the right of the base. It indicates how many times the base is used as a factor. Therefore, in the expression \(4^3\), the exponent is 3, telling us to multiply the base, 4, by itself three times.
So, remember:

• Base = the main number being multiplied.
• Exponent = how many times the multiplication occurs.

This basic understanding is crucial in solving exponentiation problems.

Multiplication

Multiplication is a fundamental concept used in exponentiation. In the context of finding values like \(4^3\), multiplication is the repetitive action performed on the base. The exponent tells us how many times to multiply the base by itself.
The process starts with the base and implements continuous multiplication as per the exponent's instructions. So for \(4^3\):

• First, multiply the base 4 by itself: \(4 \times 4 = 16\).
• Then, take the result and multiply it again by the base for as many times as the exponent instructs: \(16 \times 4 = 64\).

Each step of this process stems from understanding that multiplication is repeated addition. Knowing this allows us to systematically break down exponential expressions and find the correct product.

Calculation Steps

After understanding the roles of the base and exponent, diving into calculation steps ensures accuracy and clarity. For example, computing \(4^3\) involves a clear set of actions. They're essentially simple, but methodically laid out steps ensure the right result every time:

• **Identify the base and exponent**: Base = 4, Exponent = 3.
• **Multiply the base by itself according to exponent**: Start with \(4 \times 4 = 16\).
• **Continue Calculation**: Multiply the result by the base again: \(16 \times 4\).
• **Understand the final product**: The multiplication gives us 64.

Each step in this calculation is crucial. By systematically applying these, we translate mathematical notation into practical outcomes. Practice these steps to become comfortable with computations involving exponents.