Problem 38 - Elementary Algebra

Question

Simplify. $$ 517(35-435) $$

Step-by-Step Solution

Answer

The simplified result is -206800.

Step 1: Evaluate the Parenthesis

First, solve the expression inside the parenthesis: \(35 - 435\). Perform the subtraction to find the result: \(35 - 435 = -400\).

Step 2: Distribution

Distribute the number outside the parenthesis, which is 517, to the result from the first step. This translates to multiplying 517 by -400: \(517 \times (-400)\).

Step 3: Simplification

Multiply the numbers: \( 517 \times (-400) \). Calculate: \( 517 \times 400 = 206800 \). Since we are multiplying by -400, the result will be negative, giving us \(-206800\).

Key Concepts

Order of Operations

The order of operations is like a helpful guide that shows the sequence to follow when solving any math problem. When multiple operations are involved, it’s crucial to perform them in the correct order to get the right answer. The most widely accepted guideline is BIDMAS (or PEMDAS), which stands for Brackets/Parentheses, Indices/Exponents, Division/Multiplication, Addition/Subtraction. This order ensures that all mathematicians and students solve equations the same way.

Here’s how it works:

• Parentheses: Solve anything inside parentheses first.
• Exponents: Apply any exponents or powers.
• Multiplication and Division: Perform these next, working from left to right.
• Addition and Subtraction: Lastly, perform these operations, also from left to right.

Understanding and applying the order of operations prevents common mistakes and ensures consistent results across various problems.

Parentheses Evaluation

Parentheses play a vital role in mathematics because they dictate what operations need to be completed first. They often contain expressions that, when simplified, are used for further calculations in the broader expression. In this problem, the goal is to simplify the operation inside parentheses: \(35 - 435\).

Working out the difference:

• First, you perform the subtraction \(35 - 435\).
• The result is \(-400\), as subtracting a larger number from a smaller one results in a negative value.

This step is critical because it sets the stage for subsequent operations, such as multiplication, which we will address next.

Multiplication

Multiplication is one of the fundamental arithmetic operations and can be likened to repeated addition. In algebraic simplification, once you've sorted out any parentheses, multiplication can usually be the next step, just as in our example. Here’s how it unfolds with our problem \(517 \times (-400)\):

The process involves:

• First, you multiply 517 by the absolute value of 400, which is 517 × 400 = 206800.
• The rule with multiplication is that multiplying a positive number and a negative number results in a negative product.

Thus, the outcome of this multiplication is \(-206800\). Multiplication acts as a critical connector in simplifying algebraic expressions and converting them into simpler numerical results.

Negative Numbers

Negative numbers often trip people up, but they follow predictable rules that make dealing with them easier. Subtraction, like \(35 - 435\), often yields a negative result, especially when a larger number is subtracted from a smaller one.

Important rules for negative numbers include:

• When multiplying or dividing a positive number by a negative number, the result is always negative.
• Conversely, multiplying or dividing two negative numbers results in a positive number.

In our exercise, the multiplication \(517 \times (-400)\) results in \(-206800\). Understanding these rules aids in predicting the sign of the result during calculations, allowing you to simplify expressions effectively and avoid errors.