Problem 46 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$500(1+0.12)^{2}$$

Step-by-Step Solution

Answer

The result of the operation is 627.2.

Step 1: Solve Inside the Parentheses

According to the order of operations, we first perform calculations inside parentheses. Here we have the expression \(1 + 0.12\). So we compute this as:\[1 + 0.12 = 1.12\]

Step 2: Exponentiation

Next, we handle exponentiation, which means raising a number to a power. We have to calculate \(1.12^2\):\[1.12^2 = 1.12 \times 1.12 = 1.2544\]

Step 3: Multiplication

Finally, according to the order of operations, we perform the multiplication. We multiply the result from the exponentiation by 500.\[500 \times 1.2544 = 627.2\]

Key Concepts

Understanding Parentheses in Mathematical Expressions

In mathematics, parentheses are used to group parts of expressions that should be evaluated first. Parentheses are pivotal for indicating operations that must be executed prior to others, as dictated by the order of operations.
This rule of operation is often referred to by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When you see an expression with parentheses, your first task is to simplify the section within them.

• You perform any addition, subtraction, or other operations such as multiplication or division inside these brackets.
• For example, in the expression \(1 + 0.12\), you first add the two numbers together to get 1.12.

This process ensures that your calculations are orderly and make mathematical sense, particularly in more complex problems involving additional operations like exponentiation or fractions.

The Role of Exponentiation

After resolving the operations inside parentheses, the next step in our order of operations is dealing with exponentiation. Exponentiation involves processing powers, which are numerical expressions indicating how many times a number, known as the base, is multiplied by itself.
This is a fundamental concept in higher-level math. In the expression \(1.12^2\), here 1.12 is the base, and 2 is the exponent, indicating that you should multiply 1.12 by itself.

• So, \(1.12 \times 1.12\) equals 1.2544.
• This exponentiation step simplifies the expression further, allowing you to move on to subsequent operations such as multiplication.

It’s key to remember that exponentiation can change the scale of a number quite significantly, especially with larger bases or higher powers.

Mastering Multiplication in Expressions

Finally, multiplication comes after parentheses and exponentiation in the order of operations. Multiplication is the arithmetic operation of scaling one number by another. It's available where numbers are factors or where calculations involve multiplying the result of previous operations, such as the product from exponentiation.
In our example, after evaluating \(1.12^2\) to get 1.2544, the next step is to multiply this by 500.

• Calculate \(500 \times 1.2544\), which results in 627.2.
• This result completes the set of operations, giving you the final answer to the problem.

Multiplication plays an integral role in making sense of complex mathematical formulations as it often combines the results of simpler operations into comprehensive resolutions.