Intermediate Algebra Study Guide: Rational Expressions and Operations

Q1. Find the LCD of the rational expressions: $\frac{-7}{t}$ and $\frac{-7}{t+7}$

Background

Topic: Least Common Denominator (LCD) of Rational Expressions

This question tests your ability to find the least common denominator for rational expressions with different denominators.

Key Terms and Formulas:

• LCD: The smallest expression that is a common multiple of all denominators.

• Factor denominators if possible to help find the LCD.

Step-by-Step Guidance

1. Identify the denominators: $t$ and $t+7$.

2. Check if either denominator can be factored further (in this case, they cannot).

3. Find the LCD by multiplying the distinct denominators together.

Try solving on your own before revealing the answer!

Q2. Find the LCD of the rational expressions: $\frac{1}{4a}$ and $\frac{-7}{a^2+2a}$

Background

Topic: LCD of Rational Expressions with Polynomial Denominators

This question tests your ability to factor denominators and find the least common denominator for rational expressions.

Key Terms and Formulas:

• Factor denominators: $a^2+2a = a(a+2)$

• LCD: The smallest expression that contains all factors from each denominator.

Step-by-Step Guidance

1. Factor each denominator: $4a$ and $a^2+2a$.

2. Write $a^2+2a$ as $a(a+2)$.

3. List all unique factors from both denominators.

Try solving on your own before revealing the answer!

Q3. Find the LCD of the rational expressions: $\frac{x-8}{x^2+8x+12}$ and $\frac{x+3}{x^2-3x-10}$

Background

Topic: LCD of Rational Expressions with Quadratic Denominators

This question tests your ability to factor quadratic expressions and find the least common denominator.

Key Terms and Formulas:

• Factor quadratics: $x^2+8x+12$ and $x^2-3x-10$

• LCD: The product of all unique factors from each denominator.

Step-by-Step Guidance

1. Factor $x^2+8x+12$ and $x^2-3x-10$ into linear factors.

2. List all unique factors from both denominators.

3. Multiply the unique factors to form the LCD.

Try solving on your own before revealing the answer!

Q4. Perform the indicated operation and simplify: $\frac{3}{13x} + \frac{1}{9x}$

Background

Topic: Adding Rational Expressions with Like Denominators

This question tests your ability to add rational expressions with different coefficients but similar denominators.

Key Terms and Formulas:

• Find the LCD for $13x$ and $9x$.

• Express each fraction with the LCD as the denominator.

• Add numerators once denominators are the same.

Step-by-Step Guidance

1. Find the LCD of $13x$ and $9x$ (hint: multiply the coefficients and keep $x$).

2. Rewrite each fraction with the LCD as the denominator.

3. Add the numerators together.

Try solving on your own before revealing the answer!

Q5. Perform the indicated operation and simplify: $\frac{2}{r} + \frac{8}{r-9}$

Background

Topic: Adding Rational Expressions with Unlike Denominators

This question tests your ability to find the LCD and add rational expressions with different denominators.

Key Terms and Formulas:

• Find the LCD for $r$ and $r-9$.

• Express each fraction with the LCD as the denominator.

• Add numerators once denominators are the same.

Step-by-Step Guidance

1. Find the LCD of $r$ and $r-9$ (hint: multiply the two denominators).

2. Rewrite each fraction with the LCD as the denominator.

3. Add the numerators together.

Try solving on your own before revealing the answer!

Q6. Simplify: $\frac{\frac{x}{2}}{\frac{7}{x+6}}$

Background

Topic: Simplifying Complex Fractions

This question tests your ability to simplify a fraction where both the numerator and denominator are themselves fractions.

Key Terms and Formulas:

• Complex fraction: A fraction where the numerator and/or denominator contains a fraction.

• To simplify, multiply by the reciprocal of the denominator.

Step-by-Step Guidance

1. Rewrite the expression as $\frac{x}{2} \div \frac{7}{x+6}$.

2. Multiply by the reciprocal: $\frac{x}{2} \times \frac{x+6}{7}$.

3. Simplify the resulting expression.

Try solving on your own before revealing the answer!

Q7. Simplify: $\frac{9+\frac{3}{x}}{\frac{x}{4}+\frac{1}{12}}$

Background

Topic: Simplifying Complex Fractions with Multiple Terms

This question tests your ability to combine and simplify complex fractions with multiple terms in the numerator and denominator.

Key Terms and Formulas:

• Find a common denominator for terms in the numerator and denominator.

• Combine terms, then simplify the overall fraction.

Step-by-Step Guidance

1. Find a common denominator for $9$and$\frac{3}{x}$ in the numerator.

2. Find a common denominator for $\frac{x}{4}$ and $\frac{1}{12}$ in the denominator.

3. Rewrite the expression as a single fraction over a single fraction.

Try solving on your own before revealing the answer!

Q8. Simplify: $\frac{\frac{25y^2-9x^2}{xy}}{\frac{5}{x}-\frac{3}{y}}$

Background

Topic: Simplifying Complex Fractions with Polynomial Numerators

This question tests your ability to factor polynomials and simplify complex fractions.

Key Terms and Formulas:

• Factor $25y^2-9x^2$ as a difference of squares.

• Find a common denominator for $\frac{5}{x}$ and $\frac{3}{y}$.

• Multiply by the reciprocal to simplify.

Step-by-Step Guidance

1. Factor $25y^2-9x^2$ as $(5y-3x)(5y+3x)$.

2. Find a common denominator for $\frac{5}{x}$ and $\frac{3}{y}$.

3. Rewrite the complex fraction as a multiplication by the reciprocal.

Try solving on your own before revealing the answer!

Q9. Simplify: $\frac{m^{-1}+z^{-1}}{m^{-1}-z^{-1}}$

Background

Topic: Simplifying Expressions with Negative Exponents

This question tests your ability to rewrite negative exponents as fractions and simplify the resulting expression.

Key Terms and Formulas:

• $a^{-1} = \frac{1}{a}$

• Combine fractions in the numerator and denominator.

Step-by-Step Guidance

1. Rewrite $m^{-1}$ and $z^{-1}$ as $\frac{1}{m}$ and $\frac{1}{z}$.

2. Combine the fractions in the numerator and denominator.

3. Simplify the resulting expression.

Try solving on your own before revealing the answer!

Q10. Simplify: $(a^{-1}+b^{-1})^{-1}$

Background

Topic: Simplifying Expressions with Negative Exponents

This question tests your ability to simplify expressions with negative exponents and combine fractions.

Key Terms and Formulas:

• $a^{-1} = \frac{1}{a}$

• $(x)^{-1} = \frac{1}{x}$

• Combine fractions and apply exponent rules.

Step-by-Step Guidance

1. Rewrite $a^{-1}$ and $b^{-1}$ as $\frac{1}{a}$ and $\frac{1}{b}$.

2. Add the fractions together.

3. Apply the negative exponent to the sum.

Try solving on your own before revealing the answer!