Problem 123 - 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

\(-5^{2}\) equals -25 and \((-5)^{2}\) equals 25. The difference is due to the order of operations.

Step 1: Understand Exponentiation

An exponent refers to the number of times a number is multiplied by itself. For example, \(x^{n}\) means multiplying \(x\) by itself \(n\) times.

Step 2: Solve \(-5^{2}\)

For \(-5^{2}\), by order of operations (often remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction, from left to right), the exponent is calculated first. So, it is \( -1 \times 5^{2} = -1 \times 25 = -25 \). Thus, \(-5^{2}= -25\).

Step 3: Solve \((-5)^{2}\)

For \((-5)^{2}\), the number -5 is being squared (due to the parentheses), so it's \((-5)*(-5) = 25\). Therefore, \((-5)^{2} = 25 \).