Problem 123 - College Algebra Essentials

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

When raising a number to a power, the number is multiplied by itself the number of times indicated by the power. The difference between \(-5^{2}\) and \((-5)^{2}\) lies in the order of operations. \(-5^{2}\) equals -25, because the exponent applies only to 5, and then the negative sign is applied. \((-5)^{2}\) equals 25, because the exponent applies to -5 as a whole due to the brackets.

Step 1: Understanding Exponents

Exponents indicate how many times a number, called the 'base', is multiplied by itself. For example, \(2^{3}\) (read as 'two to the power of three') means 2 is multiplied by itself three times, which equals 8.

Step 2: Interpreting Negative Numbers with Exponents

When it comes to negative numbers and exponents, the placement of the negative sign and parentheses matters. For instance, \(-5^{2}\) and \((-5)^{2}\) have different results. This is due to the order of operations in mathematics, also known as BIDMAS or PEMDAS, which stands for Brackets, Indices (or Exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

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

In the expression \(-5^{2}\), according to the order of operations, the exponent is calculated first. Hence, we have \(5^{2}\) which equals 25. But, don't forget about the negative sign in front. So, \(-5^{2}\) is equal to -25.

Step 4: Calculate \((-5)^{2}\)

On the other hand, for the expression \((-5)^{2}\), the brackets indicate that the base for the exponent is -5. Here, the number -5 (including the negative sign) is being multiplied by itself. So, \((-5)^{2}\) equals 25.