Problem 73 - Algebra for College Students

Question

Simplify each numerical expression. $$-6(-3-9-1)$$

Step-by-Step Solution

Answer

The simplified expression is 78.

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses. Inside we have \(-3 - 9 - 1\). Begin by performing the subtraction: \(-3 - 9 = -12\), and then: \(-12 - 1 = -13\). So the expression inside the parentheses simplifies to \(-13\).

Step 2: Multiply

Now, substitute the simplified value from Step 1 back into the expression. It is \(-6(-13)\). Multiply \(-6\) by \(-13\): The multiplication of two negative numbers results in a positive number: \(-6 \times -13 = 78\).

Key Concepts

Numerical Expressions

Numerical expressions are combinations of numbers and operations that can be simplified following specific rules. In these expressions, you might see addition, subtraction, multiplication, and division, all performing specific functions. The goal is to follow the mathematical order of operations to simplify the expression.

For instance, consider \(-6(-3-9-1)\). To solve this expression, you initially look for operations within parentheses (more on that later). This expression is purely numerical, meaning it contains only numbers and basic arithmetic operations.

The simplification of a numerical expression often involves:

• Identifying and simplifying parts of the expression step by step.
• Applying arithmetic operations following the correct order.
• Ensuring that all negative numbers are correctly handled.

Ultimately, the solution to a numerical expression will provide a single number as a result, which is the case here with the answer of 78.

Parentheses in Mathematics

Parentheses in mathematics are used to indicate which operations should be performed first in an expression. They are crucial in determining the order of operations. According to the order of operations, or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), operations inside parentheses must be completed before other operations outside the parentheses.

In our example, \(-6(-3-9-1)\), the first task is to deal with the contents inside the parentheses, which is \(-3-9-1\). This means simplifying step-by-step to obtain a single number out of it.

The steps are:

• First, calculate \(-3 - 9\), resulting in \(-12\).
• Next, subtract 1 from \(-12\), resulting in \(-13\).

Once the contents of the parentheses are resolved to \(-13\), it replaces the expression inside the parentheses, allowing further multiplication with \(-6\). Parentheses are not just tools for clarity; they change the meaning of expressions and influence the final outcome.

Multiplying Negative Numbers

Multiplying negative numbers can seem tricky at first, but it's straightforward once you understand the basic rules. When you multiply two negative numbers, the result is always positive. This can be thought of as two negatives canceling each other out.

To apply this in our problem of simplifying \(-6(-13)\), we follow these steps:

• Identify both numbers in the operation: first is \(-6\), the second is \(-13\).
• Recognize that since both numbers are negative, their product will be positive.
• Multiply 6 by 13 (without considering the sign for now), which equals 78.
• This calculation gives a positive 78 because of the negative-negative multiplication rule.

So, the simplified result is \(78\). Understanding the concept of multiplying negative numbers helps prevent errors and leads to accurate results in mathematical expressions.