Problem 70 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$\left[-\frac{4}{7}-\left(-\frac{2}{5}\right)\right]\left[-\frac{3}{8}+\left(-\frac{1}{9}\right)\right]$$

Step-by-Step Solution

Answer

\(\frac{1}{12}\)

Step 1: Simplify Inside the Brackets

Start by simplifying the expressions inside the brackets. For the first bracket, \(-\frac{4}{7} - \left(-\frac{2}{5}\right)\) turns into \(-\frac{4}{7} + \frac{2}{5}\). Similarly, \(-\frac{3}{8} + \left(-\frac{1}{9}\right)\) turns into \(-\frac{3}{8} - \frac{1}{9}\).

Step 2: Perform Addition and Subtraction

Next, perform the addition and subtraction reserved in Step 1. To add or subtract fractions, they need to have the same denominator. The least common multiple (LCM) of 7 and 5 is 35. So, \(-\frac{4}{7} + \frac{2}{5}\) turns into \(-\frac{20}{35} + \frac{14}{35} = -\frac{6}{35}\). The LCM of 8 and 9 is 72. So, \(-\frac{3}{8} - \frac{1}{9}\) turns into \(-\frac{27}{72} - \frac{8}{72} = -\frac{35}{72}\).

Step 3: Perform Multiplication

Finally, perform the multiplication of the two resulting fractions from step 2. The multiplication of fractions is straightforward: \(-\frac{6}{35} \times -\frac{35}{72} = \frac{210}{2520}\). Simplifying this fraction yields \(\frac{1}{12}\).

Key Concepts

Fraction Operations

Understanding fractions becomes less scary once you learn how to work with them. Fraction operations, such as addition, subtraction, multiplication, and division, are all about making the numerators and denominators work together.
One of the most important things to remember with fractions is they must share a common denominator when adding or subtracting.
Here's a quick breakdown:

• Addition and Subtraction: Convert the fractions so they have the same denominator. This may require finding the Least Common Multiple (LCM) if they don't already match.
• Multiplication: Multiply the numerators together and then the denominators together. Simplify if possible.
• Division: Flip (or find the reciprocal of) the second fraction and multiply.

Using these rules will help keep fractions manageable as they come alive in equations and expressions. It just takes a little practice and patience!

Least Common Multiple

When dealing with fractions, especially for addition and subtraction, finding the Least Common Multiple (LCM) is crucial. The LCM is the smallest number that is a multiple of the denominators of the fractions involved.
To find the LCM:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in each list.

This number becomes the new denominator for your fractions. Converting fractions to have this common denominator might mean adjusting their numerators accordingly, maintaining the value of the fractions.
For example, with the fractions \(-\frac{4}{7}\) and \(-\frac{2}{5}\), the LCM of 7 and 5 is 35. Thus, convert these to equivalents like \(-\frac{20}{35}\) and \(+\frac{14}{35}\) before performing arithmetic operations. Mastering how to find and use the LCM is a key skill in fraction operations.

Simplification of Expressions

Simplifying expressions is the process of making them as simple as possible. This often involves combining like terms, reducing fractions, or arranging them in a cleaner, more accessible form.
In the context of fractions and the order of operations, simplification ensures that results are both precise and as straightforward as possible.
Key steps in simplification include:

• Combining Like Terms: In algebraic expressions, terms with the same variables are combined.
• Reducing Fractions: Fractions are always presented in their simplest form, meaning their numerators and denominators have no common factors other than 1.
• Applying Arithmetic Operations: Follow the proper order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Simplification of expressions not only makes your work look tidy but also ensures error-free results, making it easier to understand and communicate mathematical solutions.