Problem 103 - Introductory Algebra for College Students

Question

Describe what it means to raise a number to a power. In your description, include a discussion of the difference between \(-5^{2}\) and \((-5)^{2}\)

Step-by-Step Solution

Answer

\((-5)^{2} = 25\), because the entire expression within the parentheses is squared. \(-5^{2} = -25\), because the power operation takes precedence over the negative sign.

Step 1: Defining exponential operation

Let's begin by understanding what it means to raise a number to a power. If we have a number \(a\) and we raise it to the power \(n\), we are basically multiplying \(a\) by itself \(n\) times. For example, if we have \(2^{3}\), we are multiplying \(2\) by itself \(3\) times: \(2 * 2 * 2 = 8\).

Step 2: Dealing with negative numbers-Case 1

Now let's look at the case \(-5^{2}\). Here, it's important to recognize how precedence affects calculation. The power operation takes precedence over the negative symbol, which can be read as 'negative'. Hence, the operation is read as: negative of (\(5 * 5 = 25\)), which equals \(-25\).

Step 3: Dealing with negative numbers-Case 2

On the other hand, in \((-5)^{2}\), the negative sign is included within the parentheses. This means the entire expression within the parentheses, \(-5\), is raised to the power of \(2\). In other words, \(-5\) is multiplied by itself: \(-5 * -5 = 25\). Note the difference that a negative multiplied by a negative results in a positive number. So \((-5)^{2} = 25\).

Step 4: Summary of the crucial difference

The crucial difference in the two expressions is due to the order of operations, which assigns power operation higher precedence than negation operation. Precedence matters when a negative number and an exponent are involved.