Questions and Answers
What is the conjugate of √2 + x?
What is the conjugate of √3 + y?
What is the conjugate of 5 - √a?
What is the conjugate of 6 - √b?
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What is the conjugate of -7√5 + 8√x?
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What is the conjugate of -9√2 - 6√y?
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What is the complex conjugate of 5 - 2i?
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What is the complex conjugate of 1 + 4i?
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What is the complex conjugate of -5i?
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What is the complex conjugate of 12?
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What is the complex conjugate of 3 - 4i?
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What is the complex conjugate of 9i?
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When are complex numbers considered conjugates of each other?
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How do you perform division with complex numbers?
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What is the conjugate of (6i)?
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What is the conjugate of (1 - 2i)?
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Solve (4 - 5i) + (7 + 9i)
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Solve (6 + 12i) + (7 - 5i) - (3 - 10i)
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Solve 6i(2 - 3i)
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Solve (7 - 5i)(-3 + 10i)
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What is (√-1)?
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What is i^0?
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What is i^1?
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What is i^2?
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What is i^3?
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What is i^4?
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What is i^5?
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What is i^6?
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What is i^7?
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Simplify i^59.
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Simplify 3i^8 + 5i^15.
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What is (i^5)^7?
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Solve 3(√-10)^2.
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Simplify i^66.
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Study Notes
Algebra 2 Conjugates
- The conjugate of a binomial expression changes the sign of the second term.
- Example of a conjugate: For √2 + x, its conjugate is √2 - x.
- For expressions involving square roots:
- The conjugate of √3 + y is √3 - y.
- For 5 - √a, the conjugate is 5 + √a.
Complex Conjugates
- Complex conjugates consist of a real part that remains the same and an imaginary part that is negated.
- The complex conjugate of 5 - 2i is 5 + 2i.
- For 1 + 4i, the complex conjugate is 1 - 4i.
- The conjugate of a purely imaginary number like -5i is simply its positive counterpart, 5i.
- Real numbers are their own conjugates (e.g., the conjugate of 12 remains 12).
Properties of Complex Conjugates
- For any complex number of the form a - bi, its conjugate is a + bi.
- The conjugate of negative imaginary numbers like 9i is -9i, with the sign reversed.
- When complex numbers have the same real parts but opposite imaginary parts, they are considered conjugates.
Operations Involving Complex Numbers
- Division by complex numbers involves multiplying both numerator and denominator by the conjugate of the denominator to simplify the expression.
- Example operations:
- (4-5i) + (7+9i) simplifies to 11 + 4i.
- (6+12i) + (7-5i) - (3-10i) simplifies to 10 + 7i.
- The product of 6i and (2-3i) gives 18 + 12i.
Basic Complex Number Properties
- i represents the imaginary unit, defined as √-1.
- Powers of i cycle every four terms:
- i^0 = 1
- i^1 = i
- i^2 = -1
- i^3 = -i
- i^4 = 1 (and repeats from there).
- To simplify higher powers of i, reduce the exponent modulo 4.
Simplifications
- Simplifying i^59 results in -i since 59 mod 4 = 3.
- The expression 3i^8 + 5i^15 simplifies to 3 - 5i, with each power calculated using the properties of i.
- (i^5)^7 simplifies to -i, using the cycle of powers.
Key Reminders
- Understanding the conjugate and its properties is essential in manipulating and solving complex equations.
- Always apply conjugates when simplifying or dividing expressions involving complex numbers.
Studying That Suits You
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Description
Test your understanding of conjugates in Algebra 2 with this quiz. Each card features a mathematical expression along with its conjugate, helping reinforce your learning. Ideal for review or practice before exams.
