Problem 74 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expression simplifies to 104.

Step 1: Simplify Inside the Parentheses

First, we simplify the expression inside the parentheses by combining like terms. The expression inside is egin{equation}-3 - 4 - 6.description":"First, we simplify the expression inside the parentheses by combining like terms. The expression inside is \(-3 - 4 - 6\)." the parentheses \(-3 - 4 - 6\)." We calculate this as \(-3 - 4 - 6 = -13\).

Step 2: Remove the Parentheses

Next, remove the parentheses by dealing with the multiplication outside them. The expression simplifies to\(-8 \times (-13)\)."

Step 3: Multiply Outside the Parentheses

Now, multiply the numbers: \(-8 \times (-13)\). The multiplication of two negative numbers gives a positive result. Therefore, \((-8) \times (-13) = 104\)."

Key Concepts

Combining Like Terms

Combining like terms is an essential strategy in algebra that simplifies expressions. But what does this really mean? Simply put, it involves gathering together terms in an expression that have the same variables and powers. However, when dealing with numerical expressions like -3, -4, and -6, there are no variables to consider, we just group the numbers together.

• Look at each term. If you see numbers without variables, these are your like terms.
• Instead of variables, these numbers here are just negative numbers, so we combine them by adding them together.

In the exercise, this means adding -3, -4, and -6 so we get -13. This results from simply adding the numbers in sequence: \[-3 + (-4) = -7\] and then \[-7 + (-6) = -13\]. By simplifying to -13, we've combined these like terms into a single, manageable number.

Multiplication of Integers

Multiplying integers is fundamental in mathematics and understanding it deeply can simplify solving many problems. When we talk about multiplying integers, there are some key rules:

• If you multiply two positive numbers, the result is positive.
• If you multiply one positive and one negative number, the result is negative.
• If you multiply two negative numbers, the result is positive.

In the given exercise, after combining like terms inside the parentheses, we arrived at the expression \(-8 \times (-13)\). Here, both -8 and -13 are negative, leading to a positive outcome when multiplied together. We phrase it mathematically: \[-8 \times (-13) = 104\]. This positive result emphasizes the power of multiplying two negatives in integer multiplication.

Order of Operations

The Order of Operations is a fundamental principle in math used to determine which calculations to perform first. It follows a specific sequence often remembered by the acronym PEMDAS:

• Parentheses first
• Exponents (powers and roots, etc.)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, the focus is primarily on using the order within parentheses and then moving to multiplication. Firstly, we dealt with the expression inside the parentheses by combining \(-3 - 4 - 6\) as the order directs to handle parentheses first. This gives as -13. Then, we moved on to multiplication, acting in accordance with the PEMDAS rule by calculating \(-8 \times (-13)\). Following these steps systematically ensures that we arrive at the most simplified and correct answer.