Questions and Answers
What is the result of multiplying rational expressions A/B and C/D?
What is the result of dividing rational expressions A/B by C/D?
What is the definition of adding rational expressions?
A/D + B/D = (A+B)/D
What is the definition of subtracting rational expressions?
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What does LCD stand for?
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What are the steps for adding and subtracting rational expressions?
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What does it mean if two terms are opposites?
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What is a complex rational expression?
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What defines a complex fraction?
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How do you simplify complex fractions using division?
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How do you simplify complex fractions using the LCD?
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What is a formula?
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What is an extraneous solution?
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What is a rational equation?
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What are the steps for solving rational equations?
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What is the rate of work?
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What is the formula for distance?
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What is a ratio?
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What is the rate?
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What is a proportion?
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What is the fundamental property of proportions?
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What is a term in a proportion?
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What are extremes in a proportion?
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What are means in a proportion?
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What characterizes similar triangles?
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What is the cross product in proportions?
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What is the formula for interest?
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Study Notes
Multiplying and Dividing Rational Expressions
- Multiplying: For rational expressions A/B and C/D, the product is AC/BD.
- Dividing: For A/B divided by C/D, use A/B * D/C to get AD/BC.
Adding and Subtracting Rational Expressions
- Addition: To sum A/D and B/D, combine the numerators: (A+B)/D.
- Subtraction: For A/D minus B/D, the result is (A-B)/D.
Least Common Denominator (LCD)
- The least common denominator is the smallest multiple shared by the denominators of the rational expressions.
Steps for Adding and Subtracting Rational Expressions
- Find the LCD.
- Rewrite each expression with the LCD as the new denominator.
- Combine numerators and place over the LCD.
- Simplify the final expression if possible.
Opposite Terms
- Opposite terms share the same value but differ in sign.
Complex Rational Expressions
- These involve rational expressions in either the numerator, denominator, or both.
Complex Fractions
- Defined as fractions that include a fraction in their numerator or denominator.
Simplifying Complex Fractions via Division
- Formulate the numerator and denominator as single rational expressions.
- Perform multiplication by taking the reciprocal of the denominator.
- Simplify if applicable.
Simplifying Complex Fractions Using LCD
- Identify the LCD of all components in the complex fraction.
- Multiply the fraction by 1, expressed as LCD/LCD.
- Execute operations in the numerator and denominator ensuring no fractions remain.
- Simplify the resultant expression.
Formulas
- Equations that describe established relationships among multiple variables.
Extraneous Solutions
- Solutions that arise from the process of solving but do not satisfy the original equation.
Rational Equations
- Equations containing one or more rational expressions.
Solving Rational Equations
- Identify values that cannot be solutions.
- Multiply by the LCD to eliminate fractions.
- Solve the simplified equation.
- Confirm all solutions by substituting back into the original equation.
Rates and Ratios
- Rate of Work: Complete a job in x hours implies a completion rate of 1/x per hour.
- Distance: Computed by the formula distance = rate × time.
- Ratio: The quotient of two same-unit quantities.
- Rate: The quotient of two quantities with different units.
Proportions
- A statement that two ratios or rates are equivalent.
- Fundamental Property: If a/b = c/d, then ad = bc.
- Terms in a Proportion: Four numbers; the first and fourth are termed extremes, the second and third are means.
Similar Triangles
- Defined by corresponding sides that are proportional.
Cross Product
- In a proportion, the cross product involves multiplying the extremes and means: ab and cd.
Interest Calculation
- Interest calculated using the formula I = prt (Interest = principal × rate × time).
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Description
This quiz covers key concepts from Algebra Chapter 6, focusing on rational expressions. It includes definitions and operations such as multiplying, dividing, adding, and subtracting rational expressions as well as understanding the least common denominator (LCD). Test your knowledge and mastery of these fundamental algebraic skills.