Problem 87 - Prealgebra

Question

Use the rule for order of operations to combine the following. \(3 \cdot 2^{3}+5 \cdot 4^{2}\)

Step-by-Step Solution

Answer

The result is 104.

Step 1: Calculate Exponents

To follow the order of operations, we first need to calculate the exponents. Calculate \(2^{3}\) and \(4^{2}\). \(2^3 = 2 \times 2 \times 2 = 8\) and \(4^2 = 4 \times 4 = 16\).

Step 2: Multiply

Next, multiply the results of the exponents by their respective coefficients. \(3 \times 8 = 24\) and \(5 \times 16 = 80\).

Step 3: Add

Finally, add the two products together to get the final result. \(24 + 80 = 104\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When you see a number written as a small raised digit next to another number, that's an exponent. It tells you how many times to use the number in a multiplication. For example, in the expression \(2^3\), the number 2 is used in multiplication three times: \(2 \times 2 \times 2\). This equals 8.

This operation is crucial because it allows us to write large numbers succinctly and manage their operations efficiently.

• Identify the base and the exponent: In our example, 2 is the base and 3 is the exponent.
• Multiply the base as many times as indicated by the exponent.

When we deal with equations involving exponents, we first solve these parts since they greatly affect the rest of the math operations involved.

Multiplication

Multiplication is a mathematical operation that calculates the total of one number added to itself a certain number of times.
For instance, \(3 \times 8\) means adding 8, three times: \(8 + 8 + 8\). This is different from addition because it's repeated addition, which is much faster and necessary for solving complex problems readily.

When dealing with expressions involving multiplication after calculating exponents:

• Follow the rule "multiply before adding."
• Multiply the results from the exponent calculations by their respective coefficients.

In our exercise, after calculating the exponents as 8 and 16, we multiply them with their coefficients to get \(3 \times 8 = 24\) and \(5 \times 16 = 80\). Multiplication here turns the previous operations into meaningful numbers ready for adding up.

Addition

Addition is the process of finding the total or sum by combining two or more numbers. Once you have resolved all the exponents and multiplications, the addition step is typically last in the order of operations.
In this context, bringing the numbers together after multiplication gives the final value in the expression.

• Ensure exponents and multiplications are completed first.'
• Add the resultant numbers together.

In our example, once we arrive at the products 24 and 80, we simply add them: \(24 + 80 = 104\). This addition step is direct but essential for arriving at the correct final outcome of the problem you're solving using the order of operations.