Problem 72 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is 73.

Step 1: Multiply the First Pair of Numbers

Calculate the product of the two negative numbers: \((-7) \times (-7) = 49\). Since multiplying two negative numbers gives a positive result, the product is 49.

Step 2: Multiply the Second Pair of Numbers

Calculate the product of the negative and positive number: \((-6) \times 4 = -24\). A negative times a positive is a negative, so the product is -24.

Step 3: Subtract the Second Product from the First Product

Subtract the result from Step 2 from the result in Step 1:\(49 - (-24)\).Subtracting a negative is the same as adding the corresponding positive: \(49 + 24 = 73\).

Key Concepts

Multiplying Negative Numbers

Multiplying two negative numbers can sometimes be confusing, but it's a simple rule once you understand it. When two negative numbers are multiplied together, the result is always a positive number. This might seem counterintuitive at first. To visualize this, think about the rule "double negative equals a positive," which applies here too. When you multiply \(-7 \times -7\), each negative sign cancels the other out, leading to \(+49\). It's similar to the logic that subtracting a negative value ends up adding a positive value.
Understanding these basics can simplify your approach to solving many math problems.

Integer Operations

Integer operations involve adding, subtracting, multiplying, and dividing whole numbers, including both positive and negative numbers. When dealing with integer multiplication, like in the original exercise, the rules for signs are critical.
Here are some key points to remember:

• The product of two positive numbers is positive.
• The product of two negative numbers is positive.
• The product of a positive and a negative number is negative.

Take the expression \((-6) \times 4\), for example. Multiplying a negative by a positive number results in a negative product, which in this case is \(-24\). Remembering these rules allows you to quickly perform and simplify integer operations.

Addition and Subtraction of Integers

Adding and subtracting integers can be particularly tricky, especially when negatives are involved. The basic rule here is that subtracting a negative number is the same as adding its opposite, or equivalent positive number. In our example, you had to tackle \((49 - (-24))\).
Since subtracting a negative is effectively addition, this simplifies to \((49 + 24)\), which gives you \73\.
To perform integer addition and subtraction smoothly, consider using a number line or think in terms of moving left (subtraction) or right (addition) on the number line. This process will help you grasp the right direction for each operation.