Problem 15 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To multiply rational expressions, multiply their _____ and multiply their_____ To divide two rational expressions, multiply the first by the_____ of the second.

Step-by-Step Solution

Answer

To multiply, multiply the numerators and denominators; divide by multiplying by the reciprocal.

Step 1: Identify the operation

We are asked to multiply rational expressions. Review the process of multiplying rational expressions by understanding that we are combining two fractions.

Step 2: Determine what gets multiplied

When multiplying rational expressions, the numerators of each fraction are multiplied together as well as the denominators of each fraction.

Step 3: Understand division of rational expressions

Dividing two rational expressions involves multiplying the first expression by the reciprocal of the second. The reciprocal is obtained by swapping the numerator and denominator of the second expression.

Key Concepts

Multiplying Fractions

When multiplying fractions, whether they are simple fractions or more complex rational expressions, the process is straightforward. Each fraction has two parts: a numerator (top part) and a denominator (bottom part). To multiply two fractions:

• Multiply the numerators together. This gives the numerator of your answer.
• Multiply the denominators together. This gives the denominator of your answer.

For example, if we have the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), their product would be \( \frac{a \times c}{b \times d} \). It is vital to ensure that the fractions are simplified where possible. This means reducing the fraction to its lowest terms by finding any common factors between the numerator and the denominator.
This method applies directly to rational expressions as well, where variables can be part of the numerators and denominators. Each expression is treated as a fraction, and the same multiplication rules apply.

Dividing Fractions

Dividing fractions is a slightly different operation but is closely related to multiplication. When dividing one fraction by another, you utilize the idea of the reciprocal of a fraction. To divide fractions:

• Take the reciprocal of the divisor (the fraction you are dividing by).
• Change the division sign to multiplication.
• Multiply the first fraction by the reciprocal.

For instance, if you need to divide \( \frac{a}{b} \) by \( \frac{c}{d} \), you first find the reciprocal of \( \frac{c}{d} \), which is \( \frac{d}{c} \), and then multiply to get \( \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \).
This process is extremely useful in handling rational expressions as well, where switching an operation from division to multiplication simplifies calculations. Just like in multiplication, make sure to simplify the final expression by cancelling any common factors between the numerator and the denominator.

Reciprocal

Understanding reciprocals is crucial when dealing with division of fractions and rational expressions. A reciprocal of a fraction is another fraction where the numerator and the denominator have been swapped. In simpler terms, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
Reciprocals are handy because they allow you to convert a division problem into a multiplication problem, which is often easier to manage. When working with rational expressions, it is important to identify the reciprocal quickly:

• Swap the positions of the numerator and the denominator.
• Use this new fraction to multiply instead of divide.

Remember, the concept of a reciprocal is valid for all non-zero fractions and rational expressions. If a fraction's numerator or denominator is zero, the expression cannot have a reciprocal in the realm of real numbers.