Problem 19 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(5+6 \cdot 2\)

Step-by-Step Solution

Answer

The simplified expression is 17.

Step 1: Identify the operations

The expression given is \(5 + 6 \cdot 2\). There are addition and multiplication operations present.

Step 2: Apply the order of operations

According to the order of operations (PEMDAS/BODMAS), you must perform multiplication before addition. So calculate \(6 \cdot 2\) first.

Step 3: Multiply

Calculate \(6 \cdot 2 = 12\).

Step 4: Add the results

Now, add the result of the multiplication to \(5\). So, \(5 + 12 = 17\).

Key Concepts

PEMDAS

The term "PEMDAS" stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It is a mnemonic used to remember the order of operations in mathematics. When you come across a mathematical expression, it is crucial to perform operations in the correct sequence to get the right answer. Here's how you can apply PEMDAS:

• Parentheses: Solve anything inside parentheses first.
• Exponents: Handle exponents (such as squares or square roots) next.
• Multiplication and Division: From left to right, carry out any multiplication or division.
• Addition and Subtraction: Again, from left to right, execute any addition or subtraction.

In the given problem, we have the expression \(5 + 6 \cdot 2\). According to PEMDAS, multiplication comes before addition. Therefore, you first multiply \(6 \cdot 2\) to get 12, and then add 5, resulting in 17. Remember, following PEMDAS ensures you simplify expressions in a structured and accurate manner.

BODMAS

Similar to PEMDAS, BODMAS represents another mnemonic to help remember the order of operations. BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. It's particularly used in countries like the UK. The idea remains the same: to process mathematical expressions correctly, follow BODMAS:

• Brackets: Simplify expressions inside brackets first.
• Orders: This includes exponents and roots—any operation that deals with orders.
• Division and Multiplication: Carry these out from left to right as they appear.
• Addition and Subtraction: Perform these operations last, from left to right.

Looking at our example, \(5 + 6 \cdot 2\), and applying BODMAS, you still have to start with multiplication before moving to addition. First, calculate \(6 \cdot 2 = 12\), and then add 5 to get the final result of 17. The key to both BODMAS and PEMDAS is maintaining the specific order, ensuring expressions are simplified accurately.

Simplification

Simplification in mathematics is all about making expressions easier to work with or understand. The goal is to break down complex expressions into simpler ones without changing their values. This process helps in solving equations efficiently. Here's how you can approach simplification:

• Identify all the operations involved: Understand the different operations present, like addition or multiplication.
• Apply order of operations: Using rules like PEMDAS or BODMAS ensures everything is resolved correctly.
• Combine like terms: If there are like terms, combining them further simplifies the expression.

In the case of the expression \(5 + 6 \cdot 2\), simplification requires you to identify operations and use the correct order. First, you address multiplication \(6 \cdot 2 = 12\), then add 5, ending up with 17. Remember, simplification is about making calculations straightforward, reducing mistakes, and finding the solution efficiently. Always ensure you follow a system like PEMDAS or BODMAS as you tackle each problem.