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Chapter 1: Problem 9

Find the value of each exponential expression. $$ 7^{2} $$

Short Answer

Expert verified
The value is 49.

Step by step solution

01

Identify the Base and Exponent

The given exponential expression is \(7^{2}\). Here, the base is 7 and the exponent is 2.
02

Understand the Meaning of Exponentiation

Exponentiation means multiplying the base by itself as many times as the exponent indicates. For \(7^{2}\), it means multiplying 7 by itself 2 times.
03

Perform the Multiplication

Calculate the result of the expression: \(7 \times 7 = 49\).
04

Write the Final Answer

The value of the exponential expression \(7^{2}\) is 49.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base and Exponent
Understanding the base and exponent is vital when working with exponential expressions. In our example, given the expression \(7^{2}\), we have two components to identify: the **base** and the **exponent**.
The **base** is the number that is being multiplied. In our case, it is 7.
The **exponent** tells us how many times the base is used as a factor in the multiplication. Here, the exponent is 2.
This means we are to multiply the base, 7, by itself a total of 2 times.
Exponentiation
Exponentiation refers to the process of raising a number (the base) to a power (the exponent). It is a way to describe repeated multiplication in a concise manner.
In the expression \(7^{2}\), exponentiation helps us understand that we need to multiply 7 by itself 2 times. So, the calculation we perform is:
\[ 7 \times 7 = 49 \]
Here are some important points to remember about exponentiation:
  • When the exponent is 1, the value is the base itself (e.g., \(7^{1} = 7\)).
  • When the exponent is 0, the value is always 1, regardless of the base (e.g., \(7^{0} = 1\)).
Understanding these properties makes working with exponentiation easier.
Multiplication
Multiplication is the mathematical operation of scaling one number by another. In the context of exponentiation, it involves repeating the multiplication process.
For the expression \(7^{2}\), the multiplication is straightforward:
We multiply the base 7 by itself a total of 2 times:
\[ 7 \times 7 \]
Performing that multiplication, we find that:
\[ 7 \times 7 = 49 \]
This result shows the power of using multiplication when handling exponents. By breaking down the steps, we can easily understand and verify the results of exponential expressions.

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