Problem 78 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -8(-3-4-6) $$

Step-by-Step Solution

Answer

104

Step 1: Simplify Within the Parentheses

First, we need to work inside the parentheses. The expression inside the parentheses is \(-3 - 4 - 6\). This can be simplified by performing the operations in sequence: \(-3 - 4 = -7\), then \(-7 - 6 = -13\). So the expression inside the parentheses simplifies to \(-13\).

Step 2: Multiply the Outer Coefficient

Next, we bring down the result from the parentheses and we multiply by the coefficient outside the parentheses. This gives us the expression: \(-8(-13)\). Now, multiply \(-8\) by \(-13\), which equals \(+104\) because a negative times a negative is positive.

Key Concepts

Order of Operations

When faced with a complex numerical expression, understanding the order of operations is crucial. The order of operations is a set of rules that tells us the correct sequence to evaluate a math expression. Most commonly, this sequence is remembered by the acronym PEMDAS:

• P for Parentheses
• E for Exponents
• M for Multiplication
• D for Division
• A for Addition
• S for Subtraction

First, you handle anything inside parentheses, which ensures nested computations happen correctly. For instance, in the expression \(-8(-3-4-6)\), we prioritize the calculation within the parentheses \((-3 - 4 - 6)\).
Once we've resolved the parentheses, we proceed with other operations such as multiplication as directed by the order of operations. This approach prevents errors in simplification and ensures consistency and accuracy in calculations.

Multiplication of Integers

Multiplying integers, particularly those that are negative, requires careful attention to the rules of multiplication. In the multiplication of two integers, the result's sign depends on the signs of the numbers involved. Here’s how the sign rules work:

• Multiplying two positive numbers yields a positive result.
• Multiplying two negative numbers also results in a positive product.
• Multiplying a positive and a negative number results in a negative product.

Take the example: \(-8(-13)\).
Both numbers are negative, and according to the rule, a negative number multiplied by another negative number gives a positive result, thus yielding \(+104\).
By understanding these rules, you can multiply integers accurately regardless of the signs.

Negative Numbers

Working with negative numbers can be tricky at first, but understanding them is essential in algebra and arithmetic. Negative numbers are numbers less than zero, denoted by a minus sign \((-\)).
These numbers require special attention, especially when combined with operations like subtraction and multiplication.
For example, in the expression \((-3 - 4 - 6)\):

• Performing subtraction from a negative number means moving further left on the number line.
• Subtracting \(4\) from \(-3\) gives you \(-7\) as you move further negatives.
• Continuing with subtracting \(6\) from \(-7\) results in \(-13\).

Understanding this movement on the number line helps in simplifying more challenging expressions that involve negative numbers and mixed operations.