This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform operations (add, subtract, multiply, divide) while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). Operations work like algebra with the crucial rule: whenever you see i squared, replace it with -1! For division like (3 + 2i) divided by (1 - i), multiply numerator and denominator by the conjugate of denominator (1 + i): numerator becomes (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i + 2(-1) = 3 + 5i - 2 = 1 + 5i. Denominator: (1 - i)(1 + i) = 1 - i² = 1 - (-1) = 1 + 1 = 2. Result: (1 + 5i) ÷ 2 = 1/2 + 5i/2 = 1/2 + 5/2i. Conjugate multiplication makes denominator real! Choice A correctly performs the complex number division and expresses the result in standard form a + bi with proper handling of the conjugate method. Choice C makes a sign error in the imaginary part: the result should be 1/2 + 5/2i, not 1/2 - 5/2i—check your arithmetic when multiplying the numerator! Complex number operations summary: DIVISION—multiply top and bottom by conjugate of denominator (flip imaginary sign), simplify using (a + bi)(a - bi) = a squared + b squared (always real), then express result as a + bi by dividing both parts by denominator. These patterns are consistent! The conjugate method always produces a real denominator, making division possible.

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = -1—and dividing by multiplying by the conjugate of the denominator. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The conjugate of a - bi is a + bi, and multiplying by it makes the denominator real: for (3 + 2i)/(1 - i), use (1 + i), numerator 3 + 3i + 2i + 2i² = 3 + 5i - 2 = 1 + 5i, denominator 1 + 1 = 2, so 1/2 + (5/2)i—terrific! Choice A correctly handles the conjugate and simplifies to standard form. A distractor like choice C might flip signs incorrectly in the numerator—always compute carefully! This method works for all complex divisions—practice makes perfect. Keep going; you're gaining confidence!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform addition while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). For addition, combine real parts together and imaginary parts together: (5 + 3i) + (2 - 7i) = (5 + 2) + (3 + (-7))i = 7 + (-4)i = 7 - 4i. Choice A correctly performs the complex number addition: real parts 5 + 2 = 7, and imaginary parts 3i + (-7i) = -4i, giving 7 - 4i in standard form. Choice B incorrectly subtracts the real parts (5 - 2 = 3) instead of adding them—remember, we're adding z₁ + z₂, not subtracting! When adding complex numbers, always add real to real and imaginary to imaginary: think of it like combining like terms in algebra where 'i' is just another variable that happens to have the special property i² = -1.

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform operations (add, subtract, multiply, divide) while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). Operations work like algebra with the crucial rule: whenever you see i squared, replace it with -1! For addition and subtraction, combine real parts together and imaginary parts together: (3 + 2i) - (1 - 5i) = (3 - 1) + (2 - (-5))i = 2 + 7i. To subtract z₂ = 1 + 5i from z₁ = 6 - 2i: subtract real parts: 6 - 1 = 5, subtract imaginary parts: -2i - 5i = -7i, so z₁ - z₂ = 5 - 7i. Choice A correctly performs the complex number subtraction and expresses the result in standard form a + bi with proper handling of subtracting both parts. Choice C makes a sign error when subtracting the imaginary parts: -2i - 5i = -7i, not -5i—when subtracting positive 5i from negative 2i, you get more negative! Complex number operations summary: ADDITION/SUBTRACTION—combine real parts, combine imaginary parts: (a + bi) plus or minus (c + di) = (a plus or minus c) + (b plus or minus d)i. These patterns are consistent! Remember that subtraction distributes the negative sign to both parts of the second complex number: (6 - 2i) - (1 + 5i) = 6 - 2i - 1 - 5i.

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and the special property that a complex number times its conjugate always gives a real number. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The conjugate of z = a + bi is z̄ = a - bi, and their product is (a + bi)(a - bi) = a² - abi + abi - b²i² = a² - b²(-1) = a² + b². For z = 2 + 5i, the conjugate is z̄ = 2 - 5i, and their product is (2 + 5i)(2 - 5i) = 2² + 5² = 4 + 25 = 29. Choice B correctly calculates z × z̄ = 2² + 5² = 4 + 25 = 29, which is always a positive real number. Choice A incorrectly calculates 2² - 5² = 4 - 25 = -21, forgetting that the formula is a² + b², not a² - b², because the i² term contributes a positive value when simplified. The product of conjugates eliminates all imaginary parts, leaving only the sum of squares!

This question tests your understanding of complex numbers—numbers in form $a + bi$ where i is the imaginary unit with $i^2 = -1$—and how to find the conjugate. Complex numbers extend the real number system to include square roots of negative numbers using $i = \sqrt{-1}$, so $i^2 = -1$. The conjugate of $a + bi$ is $a - bi$, flipping the sign of the imaginary part only; for example, conjugate of $3 + 4i$ is $3 - 4i$. For $-3 + 8i$, the conjugate is $-3 - 8i$, keeping the real part the same and flipping the imaginary sign. Choice B correctly identifies the conjugate of the complex number with proper sign flip of only the imaginary part. Choice A flips the sign of the real part incorrectly: from $-3 + 8i$ to $+3 - 8i$, but the conjugate changes only the imaginary sign, not the real! Remember, conjugates are useful in division and moduli: the product of a number and its conjugate is always real and positive—keep exploring these properties, you're making great progress!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform subtraction while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). For subtraction, combine real parts and imaginary parts after distributing the negative: (6 - 2i) - (1 + 5i) = (6 - 1) + (-2 - 5)i = 5 - 7i. Choice B correctly performs the complex number subtraction and expresses the result in standard form a + bi with accurate sign handling. Choice A forgets to distribute the negative to the imaginary part: subtracting 5i means -5i, not +3i from incorrect combination—always distribute the minus sign! Complex number subtraction summary: (a + bi) - (c + di) = (a - c) + (b - d)i—these operations build your confidence in working with imaginary numbers, keep up the excellent work!

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = -1—and how to perform multiplication while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. Operations work like algebra with the crucial rule: whenever you see i squared, replace it with -1! For multiplication, use FOIL: (3 + 2i)(1 - 4i) = 31 + 3(-4i) + 2i1 + 2i(-4i) = 3 - 12i + 2i - 8i² = 3 - 10i + 8 (since -8*(-1)=8) = 11 - 10i. Choice A correctly performs the multiplication and simplifies i squared to -1, resulting in 11 - 10i. A tempting distractor like Choice C might forget to replace i squared with -1, leaving it as 3 - 10i - 8i² or mishandling signs. Always simplify i squared right away, and multiplication will be a breeze—keep up the great work!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform multiplication using FOIL while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. For multiplication, use FOIL or distribution, then simplify any i squared terms to -1. To multiply (3 + 2i)(1 - 4i): First: 3 × 1 = 3. Outer: 3 × (-4i) = -12i. Inner: 2i × 1 = 2i. Last: 2i × (-4i) = -8i². Combine: 3 - 12i + 2i - 8i² = 3 - 10i - 8i². Now replace i² with -1: 3 - 10i - 8(-1) = 3 - 10i + 8 = 11 - 10i. Choice A correctly performs the multiplication and simplifies i² = -1 to get 11 - 10i in standard form. Choice B forgets to replace i² with -1: they get 3 - 10i - 8i² but leave it as -5 - 10i instead of recognizing -8i² = -8(-1) = +8, which adds to 3 to give 11. The i² simplification is mandatory—whenever you see i², immediately replace it with -1!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform subtraction while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). For subtraction, subtract real parts and imaginary parts separately: (8 - 4i) - (3 + 9i) = (8 - 3) + (-4 - 9)i = 5 + (-13)i = 5 - 13i. Choice A correctly performs the complex number subtraction: real parts 8 - 3 = 5, and imaginary parts -4i - 9i = -13i, giving 5 - 13i in standard form. Choice D makes a sign error with the imaginary parts: when subtracting 3 + 9i, you must subtract both parts, so -4i - (+9i) = -4i - 9i = -13i, not -4i + 9i = 5i. Subtraction distributes the negative sign to both terms: z₁ - z₂ = z₁ + (-z₂) where -z₂ means flip the signs of BOTH parts!