Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify: \(-16-8 \div 4 \cdot(-2) .\) (Section \(1.8,\) Example 4 )

Step-by-Step Solution

Answer

The simplified form of the expression \(-16 - 8 \div 4 \cdot (-2)\) is \(-12\).

Step 1: Identify Operations and Their Order

First, identify all operations. In this case, we have subtraction, division, and multiplication. Remembering the order of operations (PEMDAS), division and multiplication come before subtraction. Therefore, we will perform the \(8 \div 4\) division operation and \(-2 \cdot 4\) multiplication operation first.

Step 2: Perform Division and Multiplication

Perform the division \(8 \div 4\) to get \(2\), and then the multiplication \(-2 \cdot 2\) to get \(-4\). Now our operation turns to \(-16 - -4\). The fact that there are two negatives in a row leads to elimination of the double negative, turning it into addition.

Step 3: Perform Addition

Add \(-16 + 4\) to get \(-12\). Therefore the simplified form of the expression is \(-12\).

Key Concepts

Division

Division is one of the fundamental arithmetic operations that breaks down a number into equal parts. In the order of operations, division is given priority alongside multiplication, which means it is performed before addition and subtraction. In our example, the expression involves the division of 8 by 4.

• Perform the division by taking the numerator (the number to be divided, which is 8) and dividing it by the denominator (the number you are dividing by, which is 4).
• Calculate 8 divided by 4. This results in 2, as 4 goes into 8 exactly 2 times.

Understanding division helps in simplifying complex expressions step by step. Remember, in math expressions, perform division as soon as it appears left to right in conjunction with multiplication.

Multiplication

Multiplication is another critical part of arithmetic, where we combine equal groups to find the total quantity. Similar to division, it takes precedence over subtraction. In expressions, multiplication directly follows division and is handled in a left-to-right sequence.

• Once we have the result of the division, we proceed with multiplication. In our example, this means multiplying -2 by the result of the division, which is 2.
• Calculating equation: \(-2 \times 2\) yields -4, because multiplying a positive number by a negative number results in a negative product.

Mastering multiplication ensures you can handle various algebraic expressions effectively. Always remember, multiplication is typically bundled with division in terms of precedence.

Subtraction

Subtraction is the process of taking one number away from another. It is considered lower in the hierarchy of operations and is performed after division and multiplication.

• In our example, the expression turns into a subtraction problem once the other operations are performed: it changes from \(-16 - (-4)\).
• When subtracting a negative number, it is essential to remember that the two negatives cancel each other, effectively turning the operation into addition. Hence, \(-16 - (-4)\) becomes \(-16 + 4\).
• Performing the addition results in -12, completing the simplification of the original expression.

Having a strong grasp of subtraction rules, especially when negatives are involved, helps in accurately simplifying and solving equations.