Problem 75 - Elementary Algebra

Question

For exercises \(75-78\), one part of simplifying a rational expression is completed. Problem: To simplify \(\frac{\frac{4}{15 x}}{\frac{8}{15}}\), rewrite the expression as \(\frac{4}{15 x} \div \frac{8}{15}\) and simplify. $$ \text { Incorrect Answer: } \begin{aligned} & \frac{4}{15 x} \div \frac{8}{15} \\ &=\frac{4}{15 x} \cdot \frac{15}{8} \\ &=\frac{4}{15 x} \cdot \frac{15}{4 \cdot 2} \\ &=2 x \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{2x} \).

Step 1: Rewrite the Division as Multiplication

Rewrite the division of fractions as multiplication by the reciprocal: \[ \frac{\frac{4}{15x}}{\frac{8}{15}} = \frac{4}{15x} \div \frac{8}{15} = \frac{4}{15x} \cdot \frac{15}{8} \]

Step 2: Simplify the Fractions

Multiply the numerators and denominators: \[ \frac{4 \cdot 15}{15x \cdot 8} = \frac{60}{120x} \]

Step 3: Simplify the Resulting Fraction

Divide both the numerator and the denominator by the greatest common divisor, which is 60: \[ \frac{60}{120x} = \frac{60 \div 60}{120x \div 60} = \frac{1}{2x} \]

Key Concepts

rational expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for algebra and calculus. Just like regular fractions, rational expressions can often be simplified to make them easier to work with. When simplifying rational expressions, it's important to know how to factor polynomials, find common denominators, and reduce fractions to their simplest form. For example, given the expression \(\frac{\frac{4}{15x}}{\frac{8}{15}}\), the goal is to simplify it by performing operations like multiplication or division, just as you would with numerical fractions.

reciprocal

The reciprocal of a number or expression is what you multiply that number or expression by in order to get 1. For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). When simplifying rational expressions involving division, it's often useful to convert the division into multiplication by the reciprocal. For instance, \(\frac{4}{15x} \div \frac{8}{15}\) is more easily simplified by rewriting it as \(\frac{4}{15x} \cdot \frac{15}{8}\). This process makes the expression easier to manipulate and simplify.

fraction simplification

Simplifying fractions involves reducing them to their simplest form. This means ensuring that the numerator and the denominator have no common factors other than 1. To simplify a rational expression, follow these steps:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

In our example, after converting the division into multiplication by the reciprocal, you would multiply the numerators and the denominators to get \(\frac{60}{120x}\). To simplify further, you find the GCD of 60 and 120, which is 60, and then divide both parts by 60 to get the simplest form, \(\frac{1}{2x}\).

greatest common divisor

The Greatest Common Divisor (GCD) is the largest number that divides two numbers without leaving a remainder. Finding the GCD is essential when simplifying rational expressions and fractions. For example, in the expression \(\frac{60}{120x}\), both 60 and 120 can be divided by their GCD. Here's how to find the GCD in practice:

• List the factors of each number.
• Identify the largest factor that appears in both lists.

For 60, the factors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. For 120, the factors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. The largest number that appears in both lists is 60, so the GCD is 60. Dividing the numerator and denominator of \(\frac{60}{120x}\) by 60 simplifies the fraction to \(\frac{1}{2x}\).