Problem 8 - Beginning and Intermediate Algebra

Question

Evaluate. \(8^{1}\)

Step-by-Step Solution

Answer

The value of \(8^1\) is 8.

Step 1: Understand the Expression

The expression given is \(8^1\). This expression represents '8 to the power of 1' which means you need to multiply 8 by itself 1 time. In exponential notation, \(a^n\) means 'a' multiplied by itself 'n' times.

Step 2: Apply the Power Rule

For any non-zero number 'a', \(a^1 = a\). This is because multiplying a number by itself once is the same as the number itself. Thus, applying this rule to our expression, we have \(8^1 = 8\).

Step 3: Confirm the Calculation

We see that applying the power rule where any number to the power of 1 is the number itself, we achieve the result. Therefore, \(8^1 = 8\) is indeed accurate. This confirms our evaluation.

Key Concepts

Exponential Notation

Exponential notation is a way to express numbers using bases and powers. The general form is represented as \(a^n\), where \(a\) is known as the base, and \(n\) is the exponent. This notation indicates how many times the base is multiplied by itself. For example, in \(8^1\), the base is 8 and the exponent is 1.
This concise format simplifies the representation of large numbers or when performing repeated multiplication. It plays a crucial role in advanced mathematical operations like calculus or algebra, where understanding how quantities change is essential.
Remember:

• \(a^0 = 1\), where \(a\) is any non-zero number
• \(a^1 = a\), since multiplying a number by itself once results in the number
• Exponents that are whole numbers ensure repeated multiplication of the base

Understanding exponential notation is fundamental as it simplifies and clarifies expressions in mathematics.

Power Rule

The power rule is a mathematical shortcut that simplifies solving equations involving exponents. It enables you to determine the result of raising a base to a power. Specifically, when a number is raised to the power of 1, the power rule states that the result is simply the base itself.
This is because multiplying a number by itself exactly once doesn’t change its value. For example:

• \(a^1 = a\)
• \(5^1 = 5\)
• \(8^1 = 8\)

The power rule is essential for performing operations with exponential expressions quickly and efficiently. It reduces computation by providing a straightforward outcome when the exponent is 1. Understanding this rule aids greatly in solving more complex equations as well.

Evaluate Expression

Evaluating an expression involves simplifying it to find its value. When dealing with exponents, you apply logarithmic rules and simplify accordingly. In our example, \(8^1\), evaluating means applying what we know about exponents to find the result.
Here’s how to evaluate an expression with exponents effectively:

• Identify the base and the exponent in the expression
• Apply any rules, like the power rule, to simplify
• Calculate and confirm your computation

In the given expression, applying the power rule reveals that \(8^1 = 8\). This evaluation confirms our understanding that any number raised to the power of 1 equals the number itself.
Practicing these steps will make you proficient in evaluating various expressions, making problem-solving quicker and more intuitive.