Question

1. adding, subtracting, multiplying, and dividing rational expressions are similar to performing the same operations on rational numbers. using examples for each operation, support this statement.

Explanation:

To support the statement, we analyze each operation (addition, subtraction, multiplication, division) for both rational numbers and rational expressions:

1. Addition:

• Rational Numbers: Consider \(\frac{2}{3} + \frac{4}{5}\). First, find a common denominator (15). Then, \(\frac{2\times5}{3\times5}+\frac{4\times3}{5\times3}=\frac{10}{15}+\frac{12}{15}=\frac{22}{15}\).
• Rational Expressions: Consider \(\frac{x}{y}+\frac{z}{w}\) (where \(y,w

eq0\)). The common denominator is \(yw\), so \(\frac{xw}{yw}+\frac{zy}{yw}=\frac{xw + zy}{yw}\). For example, \(\frac{1}{x}+\frac{2}{x}\) (denominator already same) gives \(\frac{1 + 2}{x}=\frac{3}{x}\), or \(\frac{1}{x}+\frac{1}{y}=\frac{y + x}{xy}\).

2. Subtraction:

• Rational Numbers: Take \(\frac{5}{6}-\frac{1}{4}\). Common denominator is 12. So, \(\frac{5\times2}{6\times2}-\frac{1\times3}{4\times3}=\frac{10}{12}-\frac{3}{12}=\frac{7}{12}\).
• Rational Expressions: Take \(\frac{3}{a}-\frac{2}{b}\) (\(a,b

eq0\)). Common denominator \(ab\), so \(\frac{3b}{ab}-\frac{2a}{ab}=\frac{3b - 2a}{ab}\). Example: \(\frac{4}{x}-\frac{1}{x}=\frac{4 - 1}{x}=\frac{3}{x}\).

3. Multiplication:

• Rational Numbers: \(\frac{3}{4}\times\frac{5}{7}=\frac{3\times5}{4\times7}=\frac{15}{28}\) (multiply numerators and denominators).
• Rational Expressions: \(\frac{m}{n}\times\frac{p}{q}=\frac{mp}{nq}\) (\(n,q

eq0\)). Example: \(\frac{x}{2}\times\frac{y}{3}=\frac{xy}{6}\).

4. Division:

• Rational Numbers: \(\frac{2}{3}\div\frac{4}{5}=\frac{2}{3}\times\frac{5}{4}=\frac{10}{12}=\frac{5}{6}\) (multiply by reciprocal).
• Rational Expressions: \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc}\) (\(b,c,d

eq0\)). Example: \(\frac{x}{y}\div\frac{z}{w}=\frac{xw}{yz}\) (e.g., \(\frac{2}{x}\div\frac{3}{y}=\frac{2y}{3x}\)).

In all cases, the process (finding common denominators for addition/subtraction, multiplying numerators/denominators for multiplication, multiplying by reciprocal for division) is analogous between rational numbers and rational expressions.

Answer:

The operations on rational expressions (addition, subtraction, multiplication, division) mirror those on rational numbers:

• Addition/Subtraction: Require a common denominator, then combine numerators (e.g., \(\frac{1}{x}+\frac{2}{x}=\frac{3}{x}\) and \(\frac{2}{3}+\frac{4}{5}=\frac{22}{15}\)).
• Multiplication: Multiply numerators and denominators (e.g., \(\frac{x}{2}\times\frac{y}{3}=\frac{xy}{6}\) and \(\frac{3}{4}\times\frac{5}{7}=\frac{15}{28}\)).
• Division: Multiply by the reciprocal (e.g., \(\frac{2}{x}\div\frac{3}{y}=\frac{2y}{3x}\) and \(\frac{2}{3}\div\frac{4}{5}=\frac{5}{6}\)).

This shows the similarity in operational logic.