Problem 44 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$6^{2}+36 \div 9$$

Step-by-Step Solution

Answer

The simplified expression is 40.

Step 1: Identify Operations

The given expression is \(6^2 + 36 \div 9\). Identify the operations involved: we have exponentiation, division, and addition.

Step 2: Apply the Order of Operations - Exponents First

According to the order of operations (PEMDAS/BODMAS), we first solve any exponents. Calculate \(6^2\):\[ 6^2 = 6 \times 6 = 36 \]

Step 3: Simplify Division

Next, perform the division \(36 \div 9\):\[ 36 \div 9 = 4 \]

Step 4: Perform Addition

Now, add the results of the exponents and division: \(36 + 4\):\[ 36 + 4 = 40 \]

Step 5: Verify the Solution

Ensure that each operation was performed correctly and in the right order. The result of the entire expression should be confirmed as 40.

Key Concepts

Understanding PEMDAS

The order of operations is crucial when simplifying expressions like \(6^2 + 36 \div 9\). PEMDAS is an acronym that stands for:

• P: Parentheses
• E: Exponents (or powers)
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

These operations must be performed in this specific sequence to ensure accurate results. This rule helps avoid ambiguity and ensures that mathematical expressions have a single, clear answer. In our example \(6^2 + 36 \div 9\), although parentheses aren't present, it's vital to follow the appropriate sequence to simplify correctly.

Exploring Exponentiation

Exponentiation is one of the first operations addressed in the order of operations, following parentheses. It involves raising a number to a power, which is a way of expressing repeated multiplication. For example, in the term \(6^2\), the number 6 is the base and 2 is the exponent.
This means 6 is multiplied by itself, leading to \(6 \times 6 = 36\). Exponentiation is key in solving expressions accurately, as it can significantly change the value of numbers when raised to powers.

Diving into Division

Division follows exponentiation in the order of operations. It is a way to determine how many times one number is contained within another. In our expression, we have \(36 \div 9\). Here, we divide 36 by 9, which equals 4.
It's crucial to perform division before moving on to addition, as dictated by PEMDAS. Properly executing division at the correct stage ensures that the expression's simplification is accurate. Remember, division and multiplication go hand-in-hand within the PEMDAS framework and are tackled from left to right.

All About Addition

Finally, addition is performed after all other operations have been completed. In the expression \(6^2 + 36 \div 9\), the addition step occurs last.

• After calculating the exponent \(6^2 = 36\)
• And dividing: \(36 \div 9 = 4\)

We now add these results together: \(36 + 4 = 40\). Addition combines all separate calculated components, resulting in the final solution. This final step completes the simplification process, confirming that all operations were executed in the correct order according to PEMDAS.