Problem 22 - Prealgebra

Question

Use the definition of exponents to expand each of the following expressions. Then multiply according to the rule for multiplication. a. \((-5)^{2}\) b. \(-5^{2}\)

Step-by-Step Solution

Answer

(a) 25, (b) -25.

Step 1: Understanding the Expression

First, we need to understand the notation given in each part. For part (a), \((-5)^2\) indicates we are squaring the entire expression inside the parentheses, which includes both the negative sign and the number. In part (b), \(-5^2\) indicates that only the 5 is squared, and then the result is negated.

Step 2: Expanding using the Definition of Exponents: Part (a)

In part (a), the expression \((-5)^2\) means \((-5) \times (-5)\). According to the rule of negative signs when multiplying, a negative times a negative results in a positive. Therefore, \((-5) \times (-5) = 25\).

Step 3: Expanding using the Definition of Exponents: Part (b)

In part (b), the expression \(-5^2\) can be written as \(-(5 \times 5)\). We first calculate the square of 5, which is \(5 \times 5 = 25\), and then we apply the negative sign, resulting in \(-25\).

Key Concepts

Understanding Negative Numbers

Negative numbers are values that are less than zero and are usually symbolized by a minus (-) sign in front of them. They are the opposite of positive numbers. It's essential to understand how to work with negative numbers because they appear frequently in mathematics, especially when dealing with operations like multiplication, division, and exponents.
When multiplying two negative numbers, the result is always positive. This might seem counterintuitive at first, but it's a consistent rule in mathematics. For instance:

• If we multiply (-3) by (-2), the result is positive 6, or (-3) × (-2) = 6.
• Negative times a negative equals a positive.
• However, when a negative number is multiplied by a positive number, the result is negative.

For example, (-4) × 5 results in -20. Understanding these rules is critical when solving problems that involve exponents and multiplication.

The Concept of Squaring

Squaring a number means multiplying the number by itself. When you square a number, you are essentially expanding the number using an exponent of 2. For example, squaring 6 is expressed as 6², which is equivalent to 6 × 6 = 36.
Interestingly, squaring negative numbers follows specific rules:

• When you square a negative number, like (-5)^2, the result is a positive number. This is because (-5) × (-5) equals 25.
• Remember that the square of any real number, whether positive or negative, is always non-negative.

However, without parentheses, squaring applies only to the number itself, and not the negative sign outside. For instance, -5^2 implies that you are squaring 5 first to get 25, and then applying the negative sign, resulting in -25.
Understanding the placement of parentheses is crucial in distinguishing between squaring a negative number and negating the square of a number.

Rules for Multiplication in Mathematics

Multiplication is one of the foundational operations in mathematics. It involves combining equal groups of numbers to find their total. In multiplication, there are specific rules, especially when dealing with signs:

• When multiplying two positive numbers, the result is always positive. For example, 4 × 3 = 12.
• If one multiplies a positive number by a negative number, the outcome is negative, such as 3 × (-2) = -6.
• Multiplying two negative numbers yields a positive result, as we saw with (-5) × (-5) = 25.

This can sometimes trip people up, particularly when dealing with expressions that include exponents, as these involve repeated multiplication. To avoid confusion, always note the position of negative signs and whether they're associated with parentheses. Proper understanding and practice of these rules ensure accuracy and clarity in problem-solving.