Problem 44 - Elementary Algebra

Question

Simplify. $$ (-3.2-3.3)(8.7-4.7)(-4.7+3.9+2.1) $$

Step-by-Step Solution

Answer

The simplified expression is -33.8.

Step 1: Simplify Inside the Parentheses

First, simplify the expressions inside the parentheses individually. For the first set of parentheses: \(-3.2 - 3.3 = -6.5\).For the second set: \(8.7 - 4.7 = 4.0\).For the third set: \(-4.7 + 3.9 + 2.1 = 1.3\).

Step 2: Multiply the Simplified Values

Now, multiply the simplified values obtained from each parenthesis: \((-6.5) \times (4.0) \times (1.3)\).

Step 3: Calculate the Product of the First Two Numbers

Calculate the product of the first two numbers:\((-6.5) \times (4.0) = -26.0\).

Step 4: Calculate the Final Product

Multiply the result from Step 3 by the remaining number:\((-26.0) \times (1.3) = -33.8\).

Key Concepts

Parentheses in Algebra

In algebra, parentheses are used to group parts of an expression. This grouping has two main purposes: to indicate that operations enclosed should be performed first, and to clarify the order of operations.
Enclosing parts of an expression in parentheses can drastically change the result.
To simplify expressions with parentheses, treat the expression inside as a separate mini-expression, and solve it completely before proceeding with other calculations.

• For instance, in the expression \((-3.2 - 3.3)\), when we see the parentheses, we first solve: \(-3.2 - 3.3 = -6.5\).
• Brackets or other types of parenthesis like \( [ ] \) and \( \{ \} \) might be used to further group expressions.

This method ensures clarity and prevents errors unrelated to solving due to misinterpretation. It’s essential when dealing with complex expressions or multiple operations.

Order of Operations

The order of operations is a set of rules to ensure that expressions are solved accurately and consistently. This is remembered through the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this order, always start simplifying expressions inside parentheses. Once all grouped values are evaluated, continue to any exponents (though our current example does not have them).
Then, proceed to multiplication or division. These are processed from left to right as they appear. In the example, after simplifying the values within parentheses, we multiply \((-6.5)\) by \(4.0\), and then use the result to multiply the next number. The PEMDAS rule is vital to avoid mistakes and guarantee that expressions are consistently simplified by everyone.

Multiplication of Real Numbers

Multiplication involves combining groups of numbers together. In mathematics, real numbers include all numbers on the number line, such as whole numbers and fractions, as well as irrational numbers like decimals.

Important points to consider when multiplying real numbers are:

• When multiplying two negative numbers, the product is positive.
• When multiplying a positive number with a negative number, the product is negative.
• The multiplication of a negative and a positive result in a negative number, as seen in our initial product: \((-6.5) \times (4.0) = -26.0\).

In algebra, careful attention to sign is key, as neglecting this can frequently lead to incorrect results.
In our example, after multiplying \(-26.0\) by \1.3\, the final product results in a negative value of \-33.8\.
Mastering the multiplication of real numbers is foundational to mastering algebraic expression simplification.