Problem 89 - Elementary Algebra

Question

\(12+4^{2} \cdot 3 \div 2+4\)

Step-by-Step Solution

Answer

40

Step 1: Evaluate the exponentiation

First, solve the exponentiation part of the expression according to the order of operations (PEMDAS/BODMAS rules). Here, calculate the value of \(4^2\). \[4^2 = 16\]

Step 2: Perform the multiplication

Next, perform the multiplication operation within the expression. Multiply the result from Step 1 by 3. \[16 \times 3 = 48\]

Step 3: Perform the division

Next, divide the product obtained in Step 2 by 2. \[48 \text{ div } 2 = 24\]

Step 4: Perform the addition

Now add all the remaining terms in the order they appear from left to right. \[12 + 24 + 4\]

Step 5: Final addition

Lastly, add the numbers obtained after each operation: \[12 + 24 + 4 = 40\]

Key Concepts

Order of Operations

Have you ever solved a math problem and got a different answer from your classmates? That's likely because the 'Order of Operations' wasn't followed. The 'Order of Operations' dictates the sequence in which mathematical operations should be performed to ensure everyone gets the same result. Quite simply, it helps us solve expressions that involve multiple operations like addition, multiplication, and parentheses. The popular acronym to remember this sequence is PEMDAS.

PEMDAS

PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

This acronym helps you remember the order. First, handle any calculations inside parentheses. Next, deal with exponents (like squares and cubes). After that, do any multiplications and divisions from left to right. Finally, perform all additions and subtractions, again from left to right. So if you encounter a problem like 12 + 4^2 * 3 / 2 + 4, the rule with PEMDAS ensures you do it correctly step by step.

Exponentiation

Exponentiation is all about raising numbers to powers. It’s expressed as a number with a tiny, raised number to its top right. Take 4^2, for instance. The 4 is the base, and the 2 is the exponent. This means 4 is multiplied by itself: 4 * 4, which equals 16. According to PEMDAS, exponentiation happens after parentheses but before multiplication, division, addition, and subtraction. So in our example: 12 + 4^2 * 3 / 2 + 4, we calculate 4^2 first, which equals 16.

Multiplication and Division

Multiplication and Division come next in the order, and they are performed from left to right. In the expression: 12 + 4^2 * 3 / 2 + 4, once we've calculated the exponent (16), next, we handle the multiplication and division. So, we multiply 16 by 3 to get 48. Then, we divide 48 by 2 to get 24. Remember, multiplication and division are treated equally in PEMDAS and performed left to right as they appear in the expression.

Addition

Addition is one of the final steps in PEMDAS. After handling parentheses, exponents, multiplication, and division, it's time to add any remaining numbers. In the expression: 12 + 4^2 * 3 / 2 + 4, after performing other operations, we end up with: 12 + 24 + 4. We simply add these from left to right to get the final answer. So, 12 + 24 is 36, and 36 + 4 is 40. Therefore, the final result is 40.