Problem 14 - College Algebra

Question

Simplify the given expression. $$ 2+8 \cdot 7 \div 4 $$

Step-by-Step Solution

Answer

The simplified expression is 16.

Step 1: Identify the Order of Operations

To simplify the expression, apply the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). There are no parentheses or exponents here, so we start with multiplication and division.

Step 2: Perform the Multiplication

First, perform the multiplication part of the expression. Calculate:\[8 \cdot 7 = 56\]Replace the multiplication in the expression with the result:\[2 + 56 \div 4\]

Step 3: Perform the Division

Next, perform the division in the expression. Calculate:\[56 \div 4 = 14\]Replace the division in the expression with the result:\[2 + 14\]

Step 4: Perform Addition

Finally, perform the addition to simplify the expression. Calculate:\[2 + 14 = 16\]Thus, the entire expression simplifies to 16.

Key Concepts

PEMDAS

PEMDAS is an acronym that helps us remember the order of operations in mathematical expressions. This order is crucial because it dictates which operations to perform first, ensuring accurate results. Each letter stands for a specific operation:

• P for Parentheses: Solve expressions within parentheses first.
• E for Exponents: Next, evaluate powers and roots.
• M for Multiplication and D for Division: Handle these operations from left to right.
• A for Addition and S for Subtraction: Finally, perform these operations from left to right.

By following PEMDAS, we can simplify expressions methodically and avoid errors that come from performing operations in the wrong order. In our example, we start with multiplication and division, as there are no parentheses or exponents involved.

simplifying expressions

Simplifying expressions means reducing them to their simplest form. This involves performing operations in the correct sequence to make the expression as straightforward as possible. Using PEMDAS is vital in this process.

Our sample expression, \(2 + 8 \cdot 7 \div 4\), may seem complex initially, but by applying PEMDAS, we can break it down into easier steps:

- Perform any multiplication or division first.

When we simplify, we break apart the expression into manageable pieces, compute each operation in order, and ultimately achieve an uncomplicated result. Simplification not only makes expressions more understandable but is also a critical skill in solving more complex problems.

multiplication and division

Multiplication and division are key steps in simplifying expressions and should always be done from left to right after parentheses and exponents have been dealt with. In our exercise, we first look at the multiplication of 8 and 7:

• Calculate \(8 \cdot 7 = 56\)

This result replaces the multiplication in our expression. Next, we tackle division:

• Calculate \(56 \div 4 = 14\)

After completing multiplication and division, the expression becomes \(2 + 14\). Both operations simplify the expression systematically, reducing complexity before moving on to addition.

addition and subtraction

Once multiplication and division are completed, we focus on addition and subtraction, moving from left to right. These are usually the final steps in the order of operations, wrapping up any remaining calculations:

In our expression, after simplifying multiplication and division, we're left with \(2 + 14\). The last operation is addition:

• Calculate \(2 + 14 = 16\)

This step finalizes the simplification process, providing the simplest form of our expression. By concluding with addition (or subtraction, if present), we ensure the expression is simplified completely. Addition and subtraction are concepts we are familiar with, making this step often the most intuitive.