Problem 99 - College Algebra

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

Raising a number to a power means multiplying it by itself a certain number of times specified by the power. The difference between \(-5^{2}\) and \((-5)^{2}\) lies in the order of operations. Without parentheses, exponentiation is performed first and negation afterwards, leading to \(-25\). With parentheses, \(-5\) is squared, resulting in \(+25\).

Step 1: Explanation of raising a number to a power

Raising a number to a power, also known as exponentiation, refers to the operation of multiplying the number by itself for the number of times specified by the power. For instance, raising 3 to the power of 2 (written as \(3^{2}\)) means multiplying 3 by itself, which equals 9.

Step 2: Difference between \(-5^{2}\) and \((-5)^{2}\)

The difference between \(-5^{2}\) and \((-5)^{2}\) lies in the order of operations. In \(-5^{2}\), the exponentiation comes first, followed by the negation. Thus, \(-5^{2}\) is computed as \(- (5 * 5)\), which equals \(-25\). On the other hand, \((-5)^{2}\) involves squaring the number \(-5\), so it is computed as \(-5 * -5\), which equals \(+25\). The parentheses change the order of the operations.