Problem 20 - College Algebra

Question

For the following exercises, simplify the given expression. $$ 64 \div(8+4 \cdot 2) $$

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Simplify Inside the Parentheses

The expression inside the parentheses is \(8 + 4 \cdot 2\). According to the order of operations, we first multiply \(4\) and \(2\), which gives us \(8\).

Step 2: Add the Result

Now, add the result from the multiplication to \(8\): \(8 + 8 = 16\). So, the expression inside the parentheses simplifies to \(16\).

Step 3: Divide

Replace the original expression with the simplified version: \(64 \div 16\). Divide \(64\) by \(16\) to get \(4\).

Key Concepts

Order of Operations

When simplifying expressions, applying the correct "order of operations" is crucial. This rule guides you in solving expressions by specifying the sequence in which operations should be performed. The well-known acronym PEMDAS helps to remember the order:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

By following PEMDAS, you ensure that each part of the mathematical expression is accurately evaluated. For example, if you encounter parentheses in an expression, the operations inside them should be solved first. Then you address exponents, followed by multiplication and division as they appear from left to right, and finally perform addition and subtraction in sequence. Ignoring this order can lead to incorrect results, showcasing why mastering this sequence is vital in algebra.

Parentheses Simplification

Parentheses serve an important purpose in mathematical expressions. They dictate which operations need to be performed first, acting as guides to prioritize calculations within them. When dealing with expressions that contain parentheses, simplifying what's inside should be your first priority.

In the given exercise, \[64 \div (8 + 4 \cdot 2)\]start by looking inside the parentheses: \[(8 + 4 \cdot 2)\]According to the order of operations, the multiplication should be performed before the addition. Begin by calculating \[4 \cdot 2 = 8\]Next, add 8 to 8:\[8 + 8 = 16\]This simplification leaves you with:\[64 \div 16\]By applying this method consistently, you ensure accurate and reliable results, making complex expressions easier to manage.

Division in Algebra

Division in algebra involves breaking down numbers or simplifying expressions even further. In the context of the example, once the expression inside the parentheses was simplified to a single number, the next step was to divide.

Division is performed as indicated in the expression, in this case:\[64 \div 16\]Dividing 64 by 16 involves determining how many times 16 can fit into 64. The result is 4, as \[64 \div 16 = 4\]When dividing in algebra, always ensure that your division is done after any necessary simplifications have been made. This step is foundational in keeping equations balanced and correctly solved. It’s also helpful to think of division as the inverse of multiplication, which can aid in checking your work and understanding the relationships between operations in algebra.