Problem 45 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$100(1+0.08)^{2}$$

Step-by-Step Solution

Answer

The result is 116.64.

Step 1: Understand the Expression

The expression is a combination of multiplication, addition, and exponentiation: \(100(1+0.08)^{2}\). Our goal is to follow the correct order of operations to solve it effectively.

Step 2: Simplify Inside the Parentheses

Start by simplifying inside the parentheses: \(1 + 0.08 = 1.08\). The expression now becomes \(100(1.08)^2\).

Step 3: Apply Exponentiation

Next, handle the exponentiation part of the expression: \((1.08)^2 = 1.1664\). Now the expression becomes \(100 \times 1.1664\).

Step 4: Multiply

Finally, multiply the remaining factors: \(100 \times 1.1664 = 116.64\). Thus, the value of the expression is 116.64.

Key Concepts

Exponentiation

Exponentiation refers to the mathematical operation where a number is raised to the power of another. This operation multiplies the number by itself a specified number of times. In our example, the number 1.08 is raised to the power of 2, which means it is multiplied by itself:

• The base is 1.08.
• The exponent is 2.
• Calculation: \((1.08)^2 = 1.08 \times 1.08 = 1.1664\).

This step is crucial because any error in exponentiation can lead to incorrect results. It's like building blocks; each power multiplies the base one more time, creating a chain reaction affecting the entire calculation. Remember, the order of operations dictates we perform exponentiation after parentheses are simplified.

Multiplication

Multiplication is a fundamental arithmetic operation where we combine equal groups or quantities together. In our expression, after resolving the exponentiation, we need to carry out multiplication:

• The number we multiply is the result from the exponentiation, which is 1.1664.
• We multiply this result by 100, reflecting an increase by a percentage factor.
• Calculation: \(100 \times 1.1664 = 116.64\).

By multiplying these, we effectively apply a scaling factor to our earlier calculations. It's essential to maintain accuracy in each step, as multiplication can greatly amplify any preceding errors, especially in financial scenarios. Multiplication helps in scaling results, which is why it often comes after simplifying and resolving all other operations within the parentheses.

Parentheses Simplification

Simplifying inside parentheses is the first step in the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). This means any calculation within parentheses needs to be addressed first before moving to other operations. In our expression:

• We have the expression \(1 + 0.08\) inside the parentheses.
• First, add these numbers: \(1 + 0.08 = 1.08\).
• The new expression then reads as \(100(1.08)^2\).

Simplifying within parentheses ensures that the rest of the operations treat the term as one single number. This method not only simplifies calculations but also ensures we follow the correct mathematical conventions, avoiding errors that might arise from processing multiple operations out of order.