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Questions and Answers
Which of the following describes how to simplify the complex fraction (3/4) / (2/5)?
Which of the following describes how to simplify the complex fraction (3/4) / (2/5)?
- Subtract 4 from 3 and divide by 2.
- Multiply 3 by 2 and 4 by 5.
- Multiply 3/4 by 5/2. (correct)
- Add 3 and 4, then divide by 2 and 5.
What is the result of adding the complex numbers (2 + 3i) and (5 + 7i)?
What is the result of adding the complex numbers (2 + 3i) and (5 + 7i)?
- 5 + 4i
- 7 + 5i
- 7 + 10i (correct)
- 3 + 10i
How do you rationalize the denominator of the expression 5/(3 + √2)?
How do you rationalize the denominator of the expression 5/(3 + √2)?
- Multiply by (3)/(3)
- Multiply by (3 + √2)/(3 + √2)
- Multiply by √2/√2
- Multiply by (3 - √2)/(3 - √2) (correct)
What is the least common denominator (LCD) of the rational expressions 1/(x + 2) and 3/(x² - 4)?
What is the least common denominator (LCD) of the rational expressions 1/(x + 2) and 3/(x² - 4)?
When multiplying the complex numbers (1 + 2i) and (3 - i), what is the result?
When multiplying the complex numbers (1 + 2i) and (3 - i), what is the result?
In the expression 2/(√5) + 3/(√5), which method is appropriate to simplify it?
In the expression 2/(√5) + 3/(√5), which method is appropriate to simplify it?
What is the method for dividing the rational expressions (x² - 9)/(x + 3) divided by (x - 3)/(2x)?
What is the method for dividing the rational expressions (x² - 9)/(x + 3) divided by (x - 3)/(2x)?
Which of the following expressions has a denominator that needs to be rationalized?
Which of the following expressions has a denominator that needs to be rationalized?
Flashcards
Complex Fractions
Complex Fractions
Fractions that contain fractions in their numerator or denominator.
Complex Numbers
Complex Numbers
A number of the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1).
Rationalizing the Denominator
Rationalizing the Denominator
The process of simplifying a fraction by removing radicals from the denominator.
Rational Expressions
Rational Expressions
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Adding/Subtracting Rational Expressions
Adding/Subtracting Rational Expressions
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Multiplying Rational Expressions
Multiplying Rational Expressions
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Dividing Rational Expressions
Dividing Rational Expressions
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Solving Rational Equations
Solving Rational Equations
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Study Notes
Complex Fractions
- Complex fractions are fractions that contain fractions in the numerator or denominator.
- To simplify complex fractions:
- Simplify the numerator and denominator separately.
- Rewrite the complex fraction as a division problem.
- Divide the simplified numerator by the simplified denominator.
- Example:
- To simplify (a/b) / (c/d), multiply the numerator by the reciprocal of the denominator.
- Thus (a/b) / (c/d) = (a/b)*(d/c) = ad/bc
Operations with Complex Numbers
- A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1).
- Operations with complex numbers:
- Addition/Subtraction: Add/subtract the real parts and the imaginary parts separately.
- Multiplication: Use the distributive property (FOIL method) and the fact that i² = -1.
- Division: Multiply the numerator and denominator by the conjugate of the denominator.
Rationalizing the Denominator
- Rationalizing the denominator is a method to eliminate radicals in a fraction's denominator.
- To rationalize a denominator with a square root:
- Multiply the numerator and denominator by the radical in the denominator to eliminate the radical.
- Example:
- To rationalize 1/√2, multiply numerator and denominator by √2 -> (1/√2) * (√2/√2) = √2/2
- If the denominator involves a sum or difference of square roots, multiply by the conjugate of the denominator.
Operations with Rational Expressions
- Expressions that can be written as a fraction where the numerator and denominator are polynomials are called rational expressions.
- Operations with rational expressions follow the same rules as with fractions:
- Addition/Subtraction: Find the least common denominator (LCD) and rewrite the fractions with the LCD. Add/subtract the numerators and put the result over the LCD.
- Multiplication: Multiply the numerators and multiply the denominators. Simplify if possible.
- Division: Multiply the first rational expression by the reciprocal of the second rational expression. Then follow the multiplication procedure.
- Solving rational equations:
- Find the least common denominator (LCD).
- Multiply both sides of the equation by the LCD.
- Simplify the result and solve the resulting equation.
Simplification Strategies
- Look for common factors in the numerator and denominator.
- Factor all polynomials to lowest terms.
- Identify the least common denominator.
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