Problem 68 - Algebra for College Students

Question

Simplify each numerical expression. $$(-7)(-7)-(-6)(4)$$

Step-by-Step Solution

Answer

The simplified expression is 73.

Step 1: Simplify the first multiplication

Start by simplifying the expression \((-7)\times(-7)\). The product of two negative numbers is positive, so calculate the multiplication: \[(-7) \times (-7) = 49\]

Step 2: Simplify the second multiplication

Now, address the expression \((-6)\times 4\). The product of a negative and a positive number is negative, so calculate the multiplication:\[(-6) \times 4 = -24\]

Step 3: Subtract the results

Combine the results of the multiplications by subtracting them according to the original expression \[49 - (-24)\]. Subtraction of a negative number is equivalent to addition, so the operation becomes:\[49 + 24 = 73\]

Step 4: Final Result

The simplified value of the expression \((-7)(-7)-(-6)(4)\) is \(73\).

Key Concepts

Multiplication of Integers

Multiplying integers might feel tricky at first, but it's quite straightforward once you know the rules. When multiplying integers, the sign of a product can help determine the outcome. Here's how it works:

• Product of two positive numbers is always positive.
• Product of two negative numbers is also positive.
• Product of a positive and a negative number is negative.

In the exercise, you had to multiply \[(-7) \times (-7)\], which uses the rule about two negative numbers resulting in a positive product. So, \[(-7) \times (-7) = 49\]. Practicing these rules helps make multiplication easier.

Negative Numbers

Negative numbers can be confusing because they act differently compared to positive numbers. Negative numbers are numbers below zero and are often used in subtracting or when you have depths, debits, or anything below a baseline.Here’s a quick guide:

• Adding a negative number is the same as subtracting its absolute value.
• Subtracting a negative number is the same as adding its absolute value.
• Negative times positive gives a negative result.

For example, in the term \((-6) \times 4\), you're multiplying a negative by a positive, which results in a negative product, hence \(-24\). Understanding these rules helps you tackle math problems with ease.

Order of Operations

Order of Operations is a crucial concept to solve problems accurately. It's about the sequence in which you perform mathematical operations. You might have heard people refer to this as PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

It’s important to follow this order to get the correct solution. In the given exercise, you first did the multiplications before handling the subtraction in\[49 - (-24)\]. Say goodbye to math mistakes by mastering this sequence.