Problem 55 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 7+8 \cdot 2 $$

Step-by-Step Solution

Answer

The simplified expression is 23.

Step 1: Identify the Operations

In the expression \(7+8 \cdot 2\), we have two operations: addition and multiplication. Based on the order of operations (PEMDAS/BODMAS), multiplication and division should be done before addition and subtraction.

Step 2: Perform the Multiplication

Execute the multiplication operation first. Multiply 8 by 2: \[8 \cdot 2 = 16\]

Step 3: Perform the Addition

After completing the multiplication, substitute 16 in its place and add 7 to get the final result:\[7 + 16 = 23\]

Key Concepts

PEMDAS

The acronym PEMDAS is a helpful tool that reminds us of the order in which we should solve operations in mathematics. It stands for:

• P: Parentheses - Solve anything inside parentheses first.
• E: Exponents - Next, any exponents or powers should be calculated.
• M: Multiplication and D: Division - Perform from left to right as they appear in the equation.
• A: Addition and S: Subtraction - Finally, perform addition and subtraction from left to right.

In the original exercise, applying PEMDAS helps us determine that multiplication is done before addition. This ensures that our calculations maintain accuracy and follow mathematical conventions.

BODMAS

BODMAS is another acronym that guides us through solving mathematical expressions, particularly used outside of the United States. It stands for:

• B: Brackets - Work on anything inside brackets first.
• O: Orders - Deal with orders, such as exponents or roots.
• D: Division and M: Multiplication - Like PEMDAS, process from left to right.
• A: Addition and S: Subtraction - Lastly, address these as they appear, left to right.

In the expression from the exercise, BODMAS tells us that multiplication is prioritized over addition. This clarifies which steps to perform first and leads to the correct answer.

Mathematical Expressions

A mathematical expression is a combination of numbers, operators, and sometimes variables, structured to show a calculation. Expressions do not have an "equals" sign, which differentiates them from equations.
Understanding the correct order of these operations is crucial. Incorrect application can lead to incorrect answers.
In the example given, the expression \(7 + 8 \cdot 2\) includes two operations. We first handle multiplication to find the product of 8 and 2. Only once that is calculated, do we proceed to addition. This ensures we discover the correct solution, 23, aligning with proper mathematical practices.