Problem 16 - Beginning and Intermediate Algebra

Question

Evaluate. \(4^{3}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Base and Exponent

The expression given is \(4^3\). Here, the base is 4, and the exponent is 3. This means that we need to multiply the number 4 by itself a total of 3 times.

Step 2: Expand the Expression

Write the expression as a multiplication of the base: \(4^3 = 4 \times 4 \times 4\). This expansion will help us perform the multiplication step-by-step.

Step 3: Multiplication of the First Pair of Numbers

First, multiply the first two 4's together: \(4 \times 4 = 16\). Keep the result to use it in the next step.

Step 4: Multiply the Result with the Last Base

Take the result from Step 3, which is 16, and multiply it by the remaining 4: \(16 \times 4 = 64\). This gives us the final result of the expression.

Key Concepts

Base and Exponent

When you come across an expression like \(4^3\), you are looking at a base and an exponent. The base is the number 4, and the exponent is 3. What this really tells you is how many times you need to multiply the base by itself.

In simple terms:

• The "base" is the number you will be multiplying.
• The "exponent" shows how many times the base will be used in the multiplication.

By understanding these two components, you can decode any exponential expression and know exactly what you need to do next.

Multiplication

Multiplication is at the heart of solving exponential expressions. For the expression \(4^3\), it requires you to multiply 4 by itself according to the number specified by the exponent.

Imagine you are organizing books on a shelf. If you have 3 of the same book, stacking them on top of each other is like multiplying that book by 3. Similarly, for \(4^3\):

• Write it as \(4 \times 4 \times 4\).
• Step 1: Multiply the first pair, \(4 \times 4 = 16\).
• Step 2: Multiply the result, 16, by the next base, 4.

This leads you to finding the final result, which is 64.

Step-by-Step Solution

Breaking down the solution into manageable steps makes it easier to understand complex expressions. Here’s a detailed walkthrough of solving \(4^3\) using a step-by-step method.

1. **Identify the Base and Exponent**: Start by recognizing \(4^3\). This helps you understand it's 4 multiplied by itself 3 times.2. **Expand the Expression**: Rewrite it as \(4 \times 4 \times 4\). This motion makes the problem more tangible.3. **First Multiplication**: Work through the first two 4's, giving \(16\).4. **Final Multiplication**: Take your result, 16, and multiply by the last 4. The final outcome of all this effort is 64.
Going through these steps teaches you to tackle similar problems methodically, ensuring accuracy and confidence in your calculations.