Problem 34 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$-5(-6-2)$$

Step-by-Step Solution

Answer

The simplified expression is 40.

Step 1: Simplify the Expression Inside the Parentheses

Start by simplifying the expression inside the parentheses: \(-6 - 2\). Subtracting 2 from -6 gives us:\[-6 - 2 = -8\]

Step 2: Apply the Multiplication Outside the Parentheses

Now that we have simplified inside the parentheses to \(-8\), multiply this by the coefficient outside, which is \(-5\). The multiplication is:\(-5 \times (-8)\). Remember, the product of two negative numbers is positive, so:\[(-5) \times (-8) = 40\]

Key Concepts

Multiplication Rules

Understanding the rules for multiplication is crucial when simplifying expressions, especially when they involve negative numbers. In multiplication, the sign of the numbers is just as important as the numbers themselves.
To multiply two numbers, follow these simple steps:

• Multiply their absolute values (disregard the sign for now). This means if you are multiplying \(-5\) and \(-8\), you first multiply 5 and 8, resulting in 40.
• Determine the sign of the product. If both numbers are negative, like in our case, the result is positive. A negative times a negative is always positive.
• If one number is negative and the other positive, the product will be negative. A positive times a negative results in a negative number.

Remember, these rules are universal in mathematics and play a vital role in ensuring accurate calculations with any equation.

Addition and Subtraction Rules

Grasping the rules for addition and subtraction helps simplify expressions, particularly when dealing with negative numbers. When you see an expression like \(-6 - 2\), what you're essentially performing is addition of negative numbers.
Here's how it works:

• Identify if the numbers have the same sign. When they do, add their absolute values and keep the sign. In \(-6 - 2\), add 6 and 2 to get 8, and keep the negative sign, resulting in \(-8\).
• If the numbers have different signs, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Understanding these operations is key to safely navigating through expressions and ensures you always land on the right answer.

Simplifying Expressions

Simplifying expressions is all about reducing them to their simplest form while strictly following the order of operations. Let's revisit the expression \(-5(-6-2)\):

• Start by solving the parentheses: \(-6 - 2 = -8\). This step often confuses students, but all you need to remember is to perform any operations inside brackets first.
• Once inside the parentheses is solved, you look outside. Here, you need to multiply \(-5\) by \(-8\), resulting in 40. Use the multiplication rules to avoid confusion with signs.

This method of simplifying expressions ensures all operations are performed properly and helps you derive the most simplified and correct answer every time.