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The distributive property is a fundamental algebra concept that allows us to simplify expressions. It helps break down more complex expressions into manageable parts. The main idea is that you can distribute, or multiply, one term across all terms inside the parentheses. The formal statement of the distributive property is:

• If you have an expression like \(a(b + c)\), it is equivalent to \(ab + ac\).

For the expression \( (8.2)(100)-(8.2)(10) \), the distributive property lets us factor out the common element, 8.2. This simplifies the work you need to do because we can rewrite the expression as \(8.2(100 - 10)\). By doing this, we're really saying: Imagine multiplying 8.2 with each term inside the parentheses rather than doing each multiplication separately.

Factoring is the process of writing an expression as a product of its factors. It is particularly useful when working with the distributive property, as it helps identify and extract common factors from an expression.

• Take a look at \(8.2(100 - 10)\).

Here, 8.2 was a common factor in both \(8.2 \times 100\) and \(8.2 \times 10\). By pulling out or factoring 8.2, we turned the original expression into a single multiplication problem. This makes the expression easier to handle, as it reduces multiple operations into a single step. Simplifying complex expressions becomes straightforward when we can identify factors to be factored out.

Arithmetic operations are basic math operations including addition, subtraction, multiplication, and division. In simplifying algebraic expressions, these operations play a vital role. Let's explore how they are used in our example.

• First, simplify within the parentheses: \((100 - 10) = 90\).
• Next, perform the multiplication: \(8.2 \times 90\).

Breaking down these individual operations helps in efficiently arriving at a solution. For this problem, once we simplified \(100 - 10\) to achieve 90, we turned our attention to calculating \(8.2 \times 90\). This could be performed manually or using a calculator for ease, yielding the final result, 738. Through clean and orderly arithmetic operations, even more complex expressions can be solved with confidence.