Problem 1 - College Algebra

Question

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

Step-by-Step Solution

Answer

No, \(2^{3} = 8\) and \(3^{2} = 9\) are different.

Step 1: Define the Powers

We start by defining the two powers given in the problem. The expression \(2^{3}\) means 2 multiplied by itself 3 times, and \(3^{2}\) means 3 multiplied by itself 2 times. We need to evaluate these expressions to compare their values.

Step 2: Calculate \(2^{3}\)

To calculate \(2^{3}\), multiply 2 by itself 3 times: \(2 \times 2 \times 2\). This can be computed as follows: \(2 \times 2 = 4\), and then \(4 \times 2 = 8\). Thus, \(2^{3} = 8\).

Step 3: Calculate \(3^{2}\)

Next, calculate \(3^{2}\), which means multiplying 3 by itself once: \(3 \times 3\). So, the calculation is \(3 \times 3 = 9\). Therefore, \(3^{2} = 9\).

Step 4: Compare Both Values

Now that we have the values of both expressions, \(2^{3} = 8\) and \(3^{2} = 9\), we compare them. Since 8 and 9 are different numbers, \(2^{3}\) is not equal to \(3^{2}\).

Key Concepts

Powers

Understanding powers is essential in multiplication of the same number repeatedly. The expression like \(2^3\) is known as a power where '2' is the base and '3' is the exponent. This means that the base, 2, is multiplied by itself three times. In general, a power is written as \(a^b\), indicating 'a' raised to the power of 'b'. It's crucial to understand that the exponent indicates how many times the base is used as a factor. Hence, \(3^2\) means 3 used as a factor twice, resulting in \(3 \times 3\). Powers help in simplifying multiplication, especially with larger numbers, because instead of multiplying over and over manually, an exponent quickly shows how many times a multiplication occurs. Powers also help in expressing very large or small numbers neatly. For instance, \(2^5\) means multiplying five 2s together, equivalent to 32. Recognizing this pattern is a step towards understanding more complex mathematical concepts and operations.

Comparing Values

Once you understand powers, comparing their results becomes straightforward. When we evaluate expressions like \(2^3\) and \(3^2\), we need to translate them into their base numbers multiplied by themselves, resulting in specific values. For \(2^3\), you multiply 2 three times, so \(2 \times 2 \times 2 = 8\). Similarly, for \(3^2\), it's \(3 \times 3 = 9\).

When comparing values, it's important to compare them numerically. Here, 8 from \(2^3\) and 9 from \(3^2\) are distinct numbers. Even though the bases and exponents are simply switched, the final values are different, teaching us that small changes in powers can lead to different outcomes.

Comparing the results of powers helps us understand the power's impact based on its base and exponent. Such comparison is a step towards deeper mathematical reasoning and problem-solving. It's a fundamental skill to predict or check results, especially for verifying computations in various mathematics applications.

Evaluation of Expressions

Evaluation of expressions involves carefully computing and simplifying mathematical phrases into a single number. In dealing with powers, evaluating expressions like \(2^3\) requires performing multiplication based on the exponent. For example, to evaluate an expression, follow these steps:

• Identify the base and the exponent.
• Perform the multiplication as per the exponent.
• Write the product as a single value.

This process is a systematic approach to breaking down powers into straightforward operations. Evaluation is essential for accuracy, ensuring that you can translate expressions into concrete numbers.

Evaluating powers goes beyond just numbers. It's about practicing the method to ensure you're correct at each step before moving forward. Once you simplify correctly, the results are trained into your memory, making more complex evaluations smooth. Always remember that while the expression is simple, correctly evaluating it strengthens analytical math skills.