Problem 5 - Algebra for College Students

Question

After reading this section, write out the answers to these questions. Use complete sentences. For which operations with rational expressions is it not necessary to have identical denominators?

Step-by-Step Solution

Answer

Identical denominators are not necessary for multiplication and division of rational expressions.

Step 1 - Understanding Rational Expressions

A rational expression is a fraction where the numerator and the denominator are polynomials. Operations include addition, subtraction, multiplication, and division.

Step 2 - Addition and Subtraction

To add or subtract rational expressions, it is necessary to have the same denominators. This is similar to adding or subtracting fractions where a common denominator is required.

Step 3 - Multiplication

For multiplication of rational expressions, it is not necessary to have identical denominators. You can multiply the numerators together and the denominators together directly.

Step 4 - Division

For division of rational expressions, it is also not necessary to have identical denominators. Instead, division is performed by multiplying by the reciprocal of the divisor.

Key Concepts

Addition and Subtraction of Rational Expressions

When dealing with the addition and subtraction of rational expressions, it is crucial to have a common denominator.
This is similar to adding or subtracting ordinary fractions. If the denominators of the rational expressions are not the same, then steps must be taken to make them identical.

Here is what you need to do to add or subtract rational expressions:

• Find the Least Common Denominator (LCD): Identify the smallest expression that both denominators can divide into.
• Rewrite the fractions: Adjust the numerators accordingly to match the LCD.
• Combine the numerators: Add or subtract the adjusted numerators.
• Simplify: Further simplify the resulting expression if possible.

Example: \(\frac{1}{x} + \frac{1}{y}\)
Find the LCD, which is \(\text{x} \times \text{y}\), leading to \(\frac{y+x}{xy}\).

Multiplication of Rational Expressions

Multiplying rational expressions is more straightforward than addition or subtraction.
It is not necessary to have identical denominators. Instead, you simply multiply across the numerators and denominators.

Follow these steps for multiplying rational expressions:

• Multiply the numerators together: Form a new numerator by multiplying the numerators of the rational expressions.
• Multiply the denominators together: Form a new denominator by multiplying the denominators of the rational expressions.
• Factor and simplify: Factor both the new numerator and the new denominator if possible, and then simplify by canceling common factors.

Example: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

Division of Rational Expressions

Division of rational expressions involves multiplying by the reciprocal of the divisor.
Like multiplication, it is not necessary to have the same denominators for division.

To divide rational expressions, use these steps:

• Rewrite the division problem: Replace the division symbol with multiplication and invert the second rational expression (take the reciprocal).
• Multiply: Use the same method as for multiplying rational expressions.
• Simplify: Factor and cancel common terms if possible.

Example: \(\frac{a}{b} \bigdiv \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\)