Problem 1 - College Algebra with Corequisite Support

Question

Is \(2^{3}\) the same as \(3^{2}\) ? Explain.

Step-by-Step Solution

Answer

No, \(2^3\) is not equal to \(3^2\) because 8 is not equal to 9.

Step 1: Understand the Problem

We need to determine if the expressions \(2^3\) and \(3^2\) are equal. This involves evaluating both expressions to see if they produce the same result.

Step 2: Evaluate \(2^3\)

Calculate \(2^3\):\[2^3 = 2 \times 2 \times 2 = 8\]. The value of \(2^3\) is 8.

Step 3: Evaluate \(3^2\)

Calculate \(3^2\):\[3^2 = 3 \times 3 = 9\]. The value of \(3^2\) is 9.

Step 4: Compare the Results

Now compare the two results we obtained: \(2^3 = 8\) and \(3^2 = 9\). Since 8 is not equal to 9, \(2^3\) is not equal to \(3^2\).

Key Concepts

Powers of Numbers

When we talk about the powers of numbers, we're discussing a type of mathematical operation called exponentiation. Exponentiation involves multiplying a number by itself a certain number of times, which is indicated by the exponent. For example, in the expression \(2^3\), 2 is the base, and 3 is the exponent. This tells us that we need to multiply 2 by itself 3 times: \(2 \times 2 \times 2\).

This results in a product of 8. On the other hand, for \(3^2\), the base is 3, and the exponent is 2, which means we multiply 3 by itself once (or \(3 \times 3\)), resulting in 9.

• The base is the number being multiplied.
• The exponent indicates how many times the base is multiplied by itself.

Understanding powers of numbers is essential because it's a cornerstone in both arithmetic and algebra, being highly useful in various mathematical problems.

Evaluating Expressions

Evaluating expressions is about finding the value of an expression by following mathematical rules and operations. In the context of powers, evaluating expressions like \(2^3\) or \(3^2\) means performing the multiplication operations that the powers indicate.

Here's how to evaluate expressions with powers:

• Identify the base and the exponent.
• Multiply the base by itself according to the exponent.
• Calculate the final product.

Taking \(2^3\) as an example, you multiply 2 by itself three times to get 8. Similarly, \(3^2\) is evaluated by multiplying 3 by itself to get 9.

Evaluating expressions correctly is vital for solving mathematical problems accurately. It ensures that you're following the right steps and understanding the relationships between numbers.

Mathematical Comparison

Mathematical comparison involves examining two expressions to determine their relationship. This could mean checking if they are equal, or if one is greater or smaller than the other. In our example, we compared \(2^3\) and \(3^2\) to see if they result in the same value.

To compare them, you evaluate each expression separately:

• Evaluate \(2^3\) and find it to be 8.
• Evaluate \(3^2\) and find it to be 9.

Once both expressions are evaluated, you can easily see that 8 and 9 are not equal, hence \(2^3\) is not the same as \(3^2\).

Understanding how to compare mathematical expressions is important because it helps in solving equations, inequalities, and problems that require logical decision-making based on numerical values.