Problem 77 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

78

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses: \(-3 - 9 - 1 = -13\)

Step 2: Multiply with the Negative Sign

Multiply the simplified value with the number outside the parentheses.\(-6 \times -13 = 78\)

Step 3: Verify the Sign

When multiplying two negative numbers, the result is positive. Therefore, the answer is correctly calculated as positive 78.

Key Concepts

Understanding Numerical Expressions

Numerical expressions are mathematical statements involving numbers and operations, such as addition, subtraction, multiplication, or division. These expressions do not contain an equality sign, unlike equations. When simplifying numerical expressions, you should follow the order of operations.
Remember the acronym PEMDAS to help guide you:

• P: Parentheses
• E: Exponents
• M/D: Multiplication/Division (from left to right)
• A/S: Addition/Subtraction (from left to right).

For example, consider the numerical expression \(-6(-3-9-1)\). Due to the presence of parentheses, inside operations should be performed first. Thus, this becomes a multi-step problem to apply different rules effectively. As you simplify each step, keep track of negative numbers and their operations.

The Role of Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign \(-\). When working with negative numbers, especially within expressions, it's important to remember the rules for addition, subtraction, multiplication, and division to get the correct results.
Here’s a simple guide to these operations:

• When adding negative numbers, think of moving left on a number line.
• Subtracting a negative number is the same as adding its positive counterpart.
• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a negative and a positive number results in a negative number.

Essentially, in the exercise, when \(-6\) was multiplied by the result inside the parentheses \(-13\), the negative signs cancel out, resulting in a positive 78.

Parentheses and Their Importance

Parentheses in math are critical for changing the normal order in which operations are carried out. By enclosing part of an expression, they indicate that these operations should be completed first.
Take the expression \(-6(-3-9-1)\). The parentheses tell us to solve \(-3-9-1\) first. The calculated value of this expression inside parentheses is then used further in operations outside them.
Without solving what is inside the parentheses first, you might incorrectly simplify the expression. Parentheses also signal grouping in mathematical expressions, crucial when dealing with negative and positive numbers together. Keeping track of operations within and outside of parentheses helps avoid mistakes, ensures proper flow, and aids in creating an accurate final result.