Problem 1 - College Algebra with Modeling and Visualization

Question

Concepts Are the expressions \(2^{3}\) and \(3^{2}\) equal? Explain your answer.

Step-by-Step Solution

Answer

No, the expressions 8 and 9 are not equal.

Step 1: Evaluate each expression

First, we evaluate the expression \(2^3\). This means 2 raised to the power of 3. When calculating \(2^3\), we multiply 2 by itself a total of 3 times: \(2 \times 2 \times 2 = 8\). Thus, \(2^3 = 8\).

Step 2: Evaluate the second expression

Next, we evaluate the expression \(3^2\). This means 3 raised to the power of 2. When calculating \(3^2\), we multiply 3 by itself a total of 2 times: \(3 \times 3 = 9\). Thus, \(3^2 = 9\).

Step 3: Compare the evaluated expressions

Now that we have evaluated both expressions, we compare the results: \(2^3 = 8\) and \(3^2 = 9\). Since 8 is not equal to 9, the expressions \(2^3\) and \(3^2\) are not equal.

Key Concepts

Powers

When we talk about powers in mathematics, we're referring to a number raised to a certain index, which tells us how many times that number is multiplied by itself. In this concept, numbers are presented in the form \(a^b\), where \(a\) is the base and \(b\) is the exponent. Understanding powers involves grasping a few key points:

• The base number (\(a\)) is the number that is being repeatedly multiplied.
• The exponent (\(b\)), also known as the power, indicates the number of times the base is used in the multiplication.
• For example, \(2^3\) means the base 2 is multiplied by itself 3 times, resulting in \(2 \times 2 \times 2 = 8\).

Using powers simplifies the representation of repeated multiplication, making it easier to calculate and understand large values. It’s essential to differentiate between bases and their respective powers, as seen in the expressions \(2^3\) and \(3^2\). Although they share similar digits, their results differ significantly.

Exponentiation

Exponentiation is the mathematical operation that involves raising a number (the base) to a power (the exponent). This operation is pivotal in various fields of mathematics and science due to its ability to easily represent large numbers that result from repeated multiplication. Here is a simple breakdown of how exponentiation works:

• The expression \(a^b\) signifies an operation where the base \(a\) is multiplied by itself \(b\) times.
• Each power has its unique value depending on both the base and the exponent.
• For instance, in the calculation of \(3^2\), we multiply 3 by itself twice: \(3 \times 3 = 9\).

Exponentiation is not just confined to whole numbers. It can also apply to fractions, decimals, and even negative and zero exponents, each with specific rules. These rules provide a framework for handling various cases, whether simplifying expressions or solving complex equations.

Mathematical Operations

Mathematical operations form the basis of arithmetic and algebra, encompassing processes like addition, subtraction, multiplication, division, and exponentiation. In understanding complex problems, breaking down expressions into simpler terms through these operations becomes vital. Consider the role of mathematical operations in evaluating powers:

• Multiplication is the fundamental operation in calculating powers, as seen in expressions like \(2^3\) where \(2\) is multiplied by itself three times.
• Comparisons also form a crucial part of operations, especially in determining the equality or inequality between different calculated powers.
• For example, once \(2^3 = 8\) and \(3^2 = 9\) are evaluated, a comparison through subtraction (or simply evaluating their equality) shows they are not equal since 8 does not equal 9.

Knowing how to apply these operations is critical for solving mathematical problems and understanding the results of expressions accurately, as they provide the tools needed to manipulate numbers and equations effectively.