Algebra Chapter 6 Flashcards

Questions and Answers

What is the result of multiplying rational expressions A/B and C/D?

AD/BC

A-B/D

AC/BD (correct)

A+B/D

What is the result of dividing rational expressions A/B by C/D?

A/B * D/C (correct)

C/D / A/B

AD/BC (correct)

A/B / 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?

A/D - B/D = (A-B)/D Signup and view all the answers

What does LCD stand for?

Least common denominator Signup and view all the answers

What are the steps for adding and subtracting rational expressions?

Find the LCD, write each expression with the LCD, add/subtract numerators, simplify. Signup and view all the answers

What does it mean if two terms are opposites?

The terms are the same but differ in sign. Signup and view all the answers

What is a complex rational expression?

A rational expression whose numerator and/or denominator contain rational expressions. Signup and view all the answers

What defines a complex fraction?

A fraction whose numerator or denominator or both contain a fraction. Signup and view all the answers

How do you simplify complex fractions using division?

Write numerator/denominator as single rational expressions, multiply by reciprocal, simplify. Signup and view all the answers

How do you simplify complex fractions using the LCD?

Find the LCD, multiply by 1 (LCD/LCD), perform operations, simplify. Signup and view all the answers

What is a formula?

An equation that states a known relationship between two or more variables. Signup and view all the answers

What is an extraneous solution?

A possible solution that does not satisfy the original equation. Signup and view all the answers

What is a rational equation?

An equation that contains one or more rational expressions. Signup and view all the answers

What are the steps for solving rational equations?

Determine invalid solutions, multiply by LCD, solve, check solutions. Signup and view all the answers

What is the rate of work?

1/x of the job is completed per hour. Signup and view all the answers

What is the formula for distance?

Distance = rate * time (r * t). Signup and view all the answers

What is a ratio?

The quotient of two numbers or quantities with the same units. Signup and view all the answers

What is the rate?

The quotient of two quantities with different units. Signup and view all the answers

What is a proportion?

A mathematical statement that two ratios or two rates are equal. Signup and view all the answers

What is the fundamental property of proportions?

If a/b = c/d, then ad = bc (and vice versa). Signup and view all the answers

What is a term in a proportion?

What each of the four numbers in a proportion are called. Signup and view all the answers

What are extremes in a proportion?

The first and fourth terms. Signup and view all the answers

What are means in a proportion?

The second and third terms. Signup and view all the answers

What characterizes similar triangles?

Triangles with corresponding sides in proportion. Signup and view all the answers

What is the cross product in proportions?

ab and cd when diagonally multiplying proportions. Signup and view all the answers

What is the formula for interest?

I = prt Signup and view all the answers

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.