Problem 70 - Introductory and Intermediate Algebra for College Students 4th

Question

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

The simplified expression is \(\frac{67}{420}\).

Step 1: Compute operations inside inner brackets

Calculate the values inside the two inner parentheses. The first parenthesis contains the operation \(- \frac{4}{7} - (- \frac{2}{5})\). Because of the double negative, this is the same as subtracting \(- \frac{4}{7}\) from \(\frac{2}{5}\), which equals - \(\frac{6}{35}\) or \(-\frac{12}{70}\). The second parenthesis contains \(- \frac{3}{8} + (- \frac{1}{9})\), which translates into subtracting \(\frac{1}{9}\) from \(- \frac{3}{8}\), which equals - \(\frac{67}{72}\). Therefore, the expression simplifies to: \[ - \frac{12}{70} \times - \frac{67}{72}\]

Step 2: Multiply the simplified fractions

Next, we simply multiply the numerators and denominators separately, which yields \(\frac{804}{5040}\) or reduced to lowest terms, \(\frac{67}{420}\).

Step 3: Apply negative sign

Finally, apply the negative sign before the fraction. But here, there are two negative signs, which make a positive. So, the product remains \(\frac{67}{420}\).

Key Concepts

Simplifying Expressions

When trying to simplify mathematical expressions, a solid understanding of the order of operations is crucial. This rule dictates the sequence in which mathematical operations should be performed to accurately simplify an expression.

Let's remember the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and finally, Addition and Subtraction (from left to right). When we simplify \(\left[-\frac{4}{7}-\left(-\frac{2}{5}\right)\right]\left[-\frac{3}{8}+\left(-\frac{1}{9}\right)\right]\), we first address the calculations inside the parentheses. In our exercise, note the presence of negative numbers and the importance of correctly interpreting the subtraction and addition of these values.

In step 1 of the solution, simplifying within the parentheses actually involves understanding that two negatives make a positive. Incorrect handling of negatives is a frequent mistake, so make sure to take extra care when simplifying expressions with negative numbers.

Fraction Arithmetic

Working with fractions can seem daunting, but it's all about being methodical. Simplifying and performing arithmetic operations with fractions involves a few main rules: finding a common denominator for addition and subtraction, and merely multiplying or dividing the numerators and denominators for multiplication and division.

In the given problem, after dealing with the inner parentheses, we have two simplified fractions ready to be multiplied. Multiplication of fractions is straightforward—multiply the numerators to get a new numerator, and do the same with the denominators. Getting \(\frac{804}{5040}\) is just the first step. To make the result more understandable, we should express it in the simplest form by dividing both the numerator and denominator by their greatest common divisor, which results in \(\frac{67}{420}\). Simplifying fractions is essential for making answers more interpretable and for allowing further operations to be carried out with ease.

Negative Numbers

Negative numbers often trip students up, particularly when combined with operations inside parentheses. A key rule to remember is that a negative number multiplied by another negative number gives a positive result.

This rule is crucial in step 3 of the problem where not one, but two negative signs are present. Instead of multiplying negatives haphazardly, recognize that the negatives cancel each other out, ultimately giving us a positive product. Learning to accurately interpret and handle negative numbers in expressions will prevent many common errors and ensures the expression simplifies correctly as seen with our final answer of \(\frac{67}{420}\), which is positive.