Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(-5(-2-6)\)

Step-by-Step Solution

Answer

The simplified form of \(-5(-2-6)\) is \(40\).

Step 1: Simplify Inside the Brackets

The expression inside the brackets is \(-2-6\). By performing the subtraction operation, \(-2 - 6 = -8\). The expression \(-5(-2-6)\) now simplifies to \(-5(-8)\).

Step 2: Perform Multiplication

After simplifying inside the brackets, now we need to multiply -5 with -8. Because multiplication of two negative numbers yields a positive number, \(-5 * -8 = 40\).

Step 3: Final Simplified Expression

Our final simplified expression is \(40\). So, \(-5(-2-6)\) simplifies to \(40\).

Key Concepts

Order of Operations

When faced with an algebraic expression like ewline{-5(-2-6)}, knowing the correct sequence to tackle the operations is crucial. This sequence is known as the order of operations, often remembered by the acronym PEMDAS in the United States or BIDMAS/BODMAS in other countries. Each letter stands for a step to be taken: Parentheses first, Exponents (i.e. powers and square roots, etc.), Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• Parentheses - Solve expressions in brackets first.
• Exponents - Evaluate powers or roots before other operations.
• Multiplication/Division - Perform these operations next, moving from left to right.
• Addition/Subtraction - Handle these last, also moving from left to right.

In our example, the operation inside the brackets, which is subtraction, comes first before we deal with the multiplication outside the brackets.

Negative Number Multiplication

Multiplying negative numbers can be a bit tricky, but there is a simple rule to remember: the product of two negative numbers is a positive number. This might seem counterintuitive at first, but it makes sense when you consider the idea that a negative number is essentially the opposite of a positive number. When you multiply two 'opposites' together, you get a positive result.

Why Negative Times Negative Equals Positive

Let's take a closer look at our example where we multiply ewline{-5} by ewline{-8}. Visualize negative multiplication as reversing direction. If ewline{-5} represents 5 steps backwards, and ewline{-8} represents reversing 8 steps (which would normally be forward), reversing a reversal gets you going forward again -- hence, a positive number. The multiplication of these two negatives according to this concept gives us a positive ewline{40}.

Algebraic Simplification

Algebraic simplification is the process of condensing an expression into its simplest form. This makes the expression easier to understand and work with. Simplification involves performing all possible operations and combining like terms.

Steps to Simplify

• Apply the order of operations: address parentheses and exponents before moving on to multiplication, division, addition, and subtraction.
• Combine like terms: group variables of the same kind and constants together.
• Simplify fractions if they are part of the expression.
• Frequently check if the expression can be factored to reduce it even further.

In the exercise ewline{-5(-2-6)}, we first addressed the parentheses by adding the two negative numbers, leading to a simplified expression of ewline{-5 * -8}. We then applied the rule for negative number multiplication, which simplified to a final answer of ewline{40}. Simplifying algebraic expressions is a fundamental skill in algebra that makes dealing with equations much more manageable.