Problem 88 - Prealgebra

Question

Use the rule for order of operations to combine the following. \(6 \cdot 5^{2}+2 \cdot 3^{3}\)

Step-by-Step Solution

Answer

The final answer is 204.

Step 1: Recognize the Order of Operations

The order of operations can be remembered using the acronym PEMDAS - Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This will guide us in solving the expression.

Step 2: Evaluate the Exponents

In the expression, identify all the exponents and solve them first. The exponents present are \(5^2\) and \(3^3\): - \(5^2 = 25\) - \(3^3 = 27\)

Step 3: Perform the Multiplications

After calculating the exponentials, perform the multiplications as follows: - Multiply \(6\) by the result of \(5^2\): \(6 \cdot 25 = 150\) - Multiply \(2\) by the result of \(3^3\): \(2 \cdot 27 = 54\)

Step 4: Add the Results

Finally, add the results of the multiplication together: \(150 + 54 = 204\)

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in mathematics that help us express repeated multiplication in a concise way. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(5^2\), 5 is the base, and the 2 is the exponent. This represents \(5 \times 5\) which equals 25. Similarly, \(3^3\) means \(3 \times 3 \times 3\), giving us 27. Exponents significantly simplify calculations that involve multiplying the same number multiple times.
Understanding exponents is crucial to applying the order of operations correctly.
When evaluating an expression like \(6 \cdot 5^{2}+2 \cdot 3^{3}\), taking care of the exponents first makes the subsequent steps simpler and ensures that the results are accurate.

Handling Multiplication in Order of Operations

Once you've addressed any exponents in an expression, the next step is to handle multiplication. Multiplication is performed after the evaluation of exponents as per the order of operations.
In our example, after calculating \(5^2\) to get 25, and \(3^3\) to get 27, we have two multiplication operations left: \(6 \cdot 25\) and \(2 \cdot 27\).
Here's how you can process this step-by-step:

• Multiply \(6\) by 25, which results in 150.
• Multiply \(2\) by 27, producing a result of 54.

Multiplying numbers after handling exponents is straightforward when you follow the correct order of operations. It ensures each part of the math problem is tackled in the right sequence, avoiding mistakes that can arise from improper operational order.

The Role of Addition in Calculation

The last step in the order of operations is addition, which is done after all exponents and multiplications have been resolved. Once you have obtained results from multiplications, like in the problem \(6 \cdot 25 = 150\) and \(2 \cdot 27 = 54\), you simply add these results together.
In our example, you should add 150 and 54:

• Add 150 + 54, which results in 204.

Addition is straightforward when done after completing other operations, but it’s crucial to ensure everything is correctly sequenced. Double-check your steps if needed. Properly following the order of operations guarantees that the addition is performed correctly, resulting in an accurate final answer.