alleryCompute (1 + i)⁸:
- -16i
- 16
- 16i
- -16
Show answer and solution
Correct answer: B. 16
(1+i) = √2 cis(π/4).
**So (1+i)⁸ = (√2)⁸ × cis(8π/4) = 16 × cis(2π) = 16 × 1 = 16.**
AP Calculus AB Unit 3B: Implicit Differentiation and Inverse Functions — practice questions — page 2
Explore 97 free Unit 3B: Implicit Differentiation and Inverse Functions questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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Showing questions 31–60 of 97.
Compute (1 + i)⁸:Read the answer and worked solution →Cube roots of unity are 1, ω, ω². Find 1 + ω + ω²:Read the answer and worked solution →Find n-th roots of unity. For z = 1: how many distinct nth roots in complex plane?Read the answer and worked solution →Find roots of z² - 4z + 5 = 0:Read the answer and worked solution →If z and z̄ are roots of a quadratic with real coefficients, then sum z + z̄ equals:Read the answer and worked solution →Locus of z satisfying |z - 1| = |z - i|:Read the answer and worked solution →The value(s) of k for which $x^2 + kx + 9 = 0$ has two equal roots is/are:Read the answer and worked solution →The roots of the equation $x^2 - 5x + 6 = 0$ are:Read the answer and worked solution →The discriminant of $2x^2 - 4x + 3 = 0$ is −8, so the equation has:Read the answer and worked solution →For a quadratic equation $ax^2+bx+c=0$, the discriminant is:Read the answer and worked solution →The standard form of a quadratic equation in the variable x is:Read the answer and worked solution →For $ax^2+bx+c=0$, the sum of the roots equals:Read the answer and worked solution →A quadratic equation has two distinct real roots when its discriminant is:Read the answer and worked solution →For $ax^2+bx+c=0$, the product of the roots equals:Read the answer and worked solution →A quadratic equation can have at most how many roots?Read the answer and worked solution →A quadratic equation has no real roots when its discriminant is:Read the answer and worked solution →A quadratic equation has two equal real roots when its discriminant is:Read the answer and worked solution →The conjugate of (2 − 3i) isRead the answer and worked solution →If z = 1 + i, then z² equalsRead the answer and worked solution →The argument (principal value) of the complex number i isRead the answer and worked solution →For complex numbers, which relation is NOT defined?Read the answer and worked solution →If z = a + ib is purely imaginary, thenRead the answer and worked solution →In z = a + ib, Re(z) and Im(z) areRead the answer and worked solution →The value of i⁴ⁿ⁺³ (n a positive integer) isRead the answer and worked solution →The modulus of the complex number 3 + 4i isRead the answer and worked solution →|z₁ z₂| equalsRead the answer and worked solution →The multiplicative inverse of (3 + 4i) isRead the answer and worked solution →The roots of x² + 1 = 0 areRead the answer and worked solution →The value of (1 + i)/(1 − i) isRead the answer and worked solution →The roots of $x^2 + 4x + 4 = 0$ are:Read the answer and worked solution →
Worked practice questions
Cube roots of unity are 1, ω, ω². Find 1 + ω + ω²:
- ω³
- 0
- 3
- 1
Show answer and solution
Correct answer: B. 0
Roots of z³ - 1 = 0 are 1, ω, ω².
Sum of roots = -coefficient of z² / coefficient of z³ = 0. So 1 + ω + ω² = 0. Useful identity for many problems.
Find n-th roots of unity. For z = 1: how many distinct nth roots in complex plane?
- Infinite
- n²
- 1
- n
Show answer and solution
Correct answer: D. n
n-th roots of unity: zk = e^(2πik/n) for k = 0, 1, ..., n-1. Exactly n distinct roots evenly spaced on the unit circle in the complex plane.
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Practise Unit 3B: Implicit Differentiation and Inverse Functions →More Mathematics topics
Unit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite LimitsUnit 1B1: Limit Concepts and Graphical Limits