Problem 13 - Beginning and Intermediate Algebra

Question

Evaluate. \(7^{2}\)

Step-by-Step Solution

Answer

\(7^{2} = 49\)

Step 1: Understanding the Exponent

The expression given is \(7^{2}\). This means that the base number, which is 7, is to be multiplied by itself once. The exponent 2 indicates the number of times you multiply the base by itself.

Step 2: Performing the Multiplication

To evaluate \(7^{2}\), multiply 7 by itself: \( 7 \times 7 \).

Step 3: Calculate the Product

Now, calculate \( 7 \times 7 \). This equals 49.

Step 4: Conclusion

Once you've done the multiplication, you have evaluated the expression entirely. Therefore, \(7^{2} = 49\).

Key Concepts

Base Number

In mathematics, the base number is fundamental when working with exponents. In the expression \(7^{2}\), the base number is 7. The base is the number that will be repeatedly multiplied when using an exponent. For instance, if the base number is 7 and the exponent is 2, as in our example, it indicates that we will multiply 7 by itself once.
Understanding the base is crucial because it serves as the starting point for any calculations involving exponents. Whether you're increasing or decreasing the exponent, it's the base that remains constant.

Multiplication

Multiplication is a key operation in mathematics that allows you to calculate repeated instances of a number. When you see an expression like \(7^{2}\), it tells you to take the base number, which is 7, and multiply it by itself as many times as indicated by the exponent. In this case, the exponent is 2, so you multiply 7 one time by itself.
Thus, it becomes \(7 \times 7\).

• Each step in multiplication involves taking two numbers and finding their combined total.
• Multiplication is essentially repeated addition.

This is straightforward yet incredibly useful when evaluating expressions with exponents.

Product

The product is the result you get from performing the multiplication operation. Within our task of evaluating \(7^{2}\), the product is the outcome of multiplying 7 by itself. Thus, the product we are looking for is 49.
Finding the product is the primary goal of a multiplication task. In our scenario:

• Multiply 7 by 7.
• This gives us the product, which is 49.

The product provides the numeric value that represents the end result of multiplying a base number by itself a certain number of times determined by the exponent.

Evaluate

To evaluate an expression means to determine its value. When given a mathematical expression like \(7^{2}\), evaluating it involves using each of the previous concepts to find the final answer. The process includes:

• Identifying the base number, which is 7.
• Recognizing the exponent, which shows how many times to multiply the base by itself (2 times in this exercise).
• Performing the multiplication \(7 \times 7\).

This gives us a product of 49.
Thus, evaluating \(7^{2}\) means finding that it equals 49, completing the expression's evaluation. The method provides a way to translate mathematical notation into meaningful numerical results.