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Algebra 2 Flashcards: Operations With Complex Numbers

Study Operations With Complex Numbers in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra 2

Operations With Complex Numbers

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QUESTION
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What is (10−3i)−(4+9i)(10-3i)-(4+9i) in standard form?

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ANSWER

6−12i6-12i. Subtract real parts: 10−4=610-4=6, imaginary parts: −3−9=−12-3-9=-12.

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What this deck covers

This deck focuses on Operations With Complex Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: What is (10−3i)−(4+9i)(10-3i)-(4+9i) in standard form?

Answer: 6−12i6-12i. Subtract real parts: 10−4=610-4=6, imaginary parts: −3−9=−12-3-9=-12.

Flashcard 2: What is (2i)(5−4i)(2i)(5-4i) in standard form?

Answer: 8+10i8+10i. Distribute: (2i)(5)+(2i)(−4i)=10i+8i2=10i−8(2i)(5) + (2i)(-4i) = 10i + 8i^2 = 10i - 8.

Flashcard 3: What is (5−2i)(1+3i)(5-2i)(1+3i) in standard form?

Answer: 11+13i11+13i. Use FOIL: (5)(1)+(5)(3i)+(−2i)(1)+(−2i)(3i)=5+15i−2i+6(5)(1) + (5)(3i) + (-2i)(1) + (-2i)(3i) = 5 + 15i - 2i + 6.

Flashcard 4: What is (7i)(−3i)(7i)(-3i) simplified?

Answer: 2121. Since (7i)(−3i)=−21i2=−21(−1)=21(7i)(-3i) = -21i^2 = -21(-1) = 21.

Flashcard 5: What is the imaginary part of the complex number a+bia+bi?

Answer: bb. The coefficient of ii in standard form.

Flashcard 6: What is the value of i5i^5 written in simplest form?

Answer: ii. Since i5=i4⋅i=1⋅i=ii^5 = i^4 \cdot i = 1 \cdot i = i.

Flashcard 7: What is the value of i2i^2?

Answer: −1-1. By definition of the imaginary unit.

Flashcard 8: What is (2+i)(3+4i)(2+i)(3+4i) in standard form?

Answer: 2+11i2+11i. Use FOIL: (2)(3)+(2)(4i)+(i)(3)+(i)(4i)=6+8i+3i−4(2)(3) + (2)(4i) + (i)(3) + (i)(4i) = 6 + 8i + 3i - 4.

Flashcard 9: What is (−5i)+(12i)(-5i)+(12i) in standard form?

Answer: 7i7i. Add imaginary parts: −5i+12i=7i-5i + 12i = 7i.

Flashcard 10: What is the multiplicative identity for complex numbers?

Answer: 1+0i1+0i. Multiplying any complex number by this yields the same number.

Flashcard 11: What is (3+2i)+(5−7i)(3+2i)+(5-7i) in standard form?

Answer: 8−5i8-5i. Add real parts: 3+5=83+5=8, imaginary parts: 2+(−7)=−52+(-7)=-5.

Flashcard 12: What is (1+i)(1−i)(1+i)(1-i) in standard form?

Answer: 22. Difference of squares: (1)2−(i)2=1−i2=1+1=2(1)^2 - (i)^2 = 1 - i^2 = 1 + 1 = 2.

Flashcard 13: What is (−3+4i)−(8+i)(-3+4i)-(8+i) in standard form?

Answer: −11+3i-11+3i. Subtract real parts: −3−8=−11-3-8=-11, imaginary parts: 4−1=34-1=3.

Flashcard 14: What is the standard form of a complex number?

Answer: a+bia+bi. Where aa and bb are real numbers and ii is the imaginary unit.

Flashcard 15: What is (a+bi)(a−bi)(a+bi)(a-bi) simplified using i2=−1i^2=-1?

Answer: a2+b2a^2+b^2. Conjugate multiplication yields sum of squares.

Flashcard 16: What is (4−9i)+(−6+3i)(4-9i)+(-6+3i) in standard form?

Answer: −2−6i-2-6i. Add real parts: 4+(−6)=−24+(-6)=-2, imaginary parts: −9+3=−6-9+3=-6.

Flashcard 17: What is (3−2i)(−1+i)(3-2i)(-1+i) in standard form?

Answer: −1+5i-1+5i. Use FOIL: (3)(−1)+(3)(i)+(−2i)(−1)+(−2i)(i)=−3+3i+2i+2(3)(-1) + (3)(i) + (-2i)(-1) + (-2i)(i) = -3 + 3i + 2i + 2.

Flashcard 18: What is (2−3i)2(2-3i)^2 in standard form?

Answer: −5−12i-5-12i. Square using (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2 and i2=−1i^2 = -1.

Flashcard 19: What is (2+5i)(−3−2i)(2+5i)(-3-2i) in standard form?

Answer: 4−19i4-19i. Use FOIL: (2)(−3)+(2)(−2i)+(5i)(−3)+(5i)(−2i)=−6−4i−15i−10(2)(-3) + (2)(-2i) + (5i)(-3) + (5i)(-2i) = -6 - 4i - 15i - 10.

Flashcard 20: What is (3+i)2(3+i)^2 in standard form?

Answer: 8+6i8+6i. Square using (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 and i2=−1i^2 = -1.

Flashcard 21: What is (bi)(di)(bi)(di) simplified using i2=−1i^2=-1?

Answer: −bd-bd. Since (bi)(di)=bdi2=bd(−1)=−bd(bi)(di) = bdi^2 = bd(-1) = -bd.

Flashcard 22: What is (2+3i)+(2−3i)(2+3i)+(2-3i) in standard form?

Answer: 44. The imaginary parts cancel: 2+2=42+2=4, 3i+(−3i)=03i+(-3i)=0.

Flashcard 23: What is the value of i4i^4 written in simplest form?

Answer: 11. Since i4=(i2)2=(−1)2=1i^4 = (i^2)^2 = (-1)^2 = 1.

Flashcard 24: What is 2(3−5i)2(3-5i) simplified to standard form?

Answer: 6−10i6-10i. Distribute: 2⋅3=62 \cdot 3 = 6, 2⋅(−5i)=−10i2 \cdot (-5i) = -10i.

Flashcard 25: What is (4+3i)(4−3i)(4+3i)(4-3i) in standard form?

Answer: 2525. Difference of squares: (4)2−(3i)2=16−9i2=16+9=25(4)^2 - (3i)^2 = 16 - 9i^2 = 16 + 9 = 25.

Flashcard 26: What is (1−2i)(−4+3i)(1-2i)(-4+3i) in standard form?

Answer: 2+11i2+11i. Use FOIL: (1)(−4)+(1)(3i)+(−2i)(−4)+(−2i)(3i)=−4+3i+8i+6(1)(-4) + (1)(3i) + (-2i)(-4) + (-2i)(3i) = -4 + 3i + 8i + 6.

Flashcard 27: What is (3+4i)(−3+4i)(3+4i)(-3+4i) in standard form?

Answer: −25-25. Use FOIL: (3)(−3)+(3)(4i)+(4i)(−3)+(4i)(4i)=−9+12i−12i−16=−25(3)(-3) + (3)(4i) + (4i)(-3) + (4i)(4i) = -9 + 12i - 12i - 16 = -25.

Flashcard 28: What is the additive identity for complex numbers?

Answer: 0+0i0+0i. Adding this to any complex number yields the same number.

Flashcard 29: What is the product (a+bi)(c+di)(a+bi)(c+di) in standard form?

Answer: (ac−bd)+(ad+bc)i(ac-bd)+(ad+bc)i. Use FOIL and i2=−1i^2 = -1 to simplify.

Flashcard 30: What is (6+i)(−2+3i)(6+i)(-2+3i) in standard form?

Answer: −15+16i-15+16i. Use FOIL: (6)(−2)+(6)(3i)+(i)(−2)+(i)(3i)=−12+18i−2i−3(6)(-2) + (6)(3i) + (i)(-2) + (i)(3i) = -12 + 18i - 2i - 3.

Flashcard 31: What is (1+i)(1+i)(1+i)(1+i) in standard form?

Answer: 2i2i. Use FOIL: (1)(1)+(1)(i)+(i)(1)+(i)(i)=1+i+i−1=2i(1)(1) + (1)(i) + (i)(1) + (i)(i) = 1 + i + i - 1 = 2i.

Flashcard 32: What is (a+bi)+(c−di)(a+bi)+(c-di) simplified in standard form?

Answer: (a+c)+(b−d)i(a+c)+(b-d)i. Add real parts and subtract imaginary parts.

Flashcard 33: What is the difference (a+bi)−(c+di)(a+bi)-(c+di) in standard form?

Answer: (a−c)+(b−d)i(a-c)+(b-d)i. Subtract real parts and imaginary parts separately.

Flashcard 34: What is −4(−2+3i)-4(-2+3i) simplified to standard form?

Answer: 8−12i8-12i. Distribute: −4⋅(−2)=8-4 \cdot (-2) = 8, −4⋅3i=−12i-4 \cdot 3i = -12i.

Flashcard 35: What is (7+i)−(2−5i)(7+i)-(2-5i) in standard form?

Answer: 5+6i5+6i. Subtract real parts: 7−2=57-2=5, imaginary parts: 1−(−5)=61-(-5)=6.

Flashcard 36: What is the value of i3i^3 written in simplest form?

Answer: −i-i. Since i3=i2⋅i=(−1)⋅i=−ii^3 = i^2 \cdot i = (-1) \cdot i = -i.

Flashcard 37: What is the sum (a+bi)+(c+di)(a+bi)+(c+di) in standard form?

Answer: (a+c)+(b+d)i(a+c)+(b+d)i. Add real parts and imaginary parts separately.

Flashcard 38: What is (1+6i)(1−6i)(1+6i)(1-6i) in standard form?

Answer: 3737. Difference of squares: (1)2−(6i)2=1−36i2=1+36=37(1)^2 - (6i)^2 = 1 - 36i^2 = 1 + 36 = 37.

Flashcard 39: What is (2+i)(2−i)(2+ i)(2- i) in standard form?

Answer: 55. Difference of squares: (2)2−(i)2=4−i2=4+1=5(2)^2 - (i)^2 = 4 - i^2 = 4 + 1 = 5.

Flashcard 40: What is the value of i6i^6 written in simplest form?

Answer: −1-1. Since i6=(i2)3=(−1)3=−1i^6 = (i^2)^3 = (-1)^3 = -1.

Flashcard 41: What is (a+bi)−(c−di)(a+bi)-(c-di) simplified in standard form?

Answer: (a−c)+(b+d)i(a-c)+(b+d)i. Distribute the negative sign to both terms inside parentheses.

Flashcard 42: What is (9−i)+(−9+i)(9- i)+( -9+ i) in standard form?

Answer: 00. The real and imaginary parts are additive inverses.

Flashcard 43: What is the real part of the complex number a+bia+bi?

Answer: aa. The coefficient of the non-imaginary term.

Flashcard 44: What is (2−i)(−5−i)(2- i)(-5- i) in standard form?

Answer: −11+3i-11+3i. Use FOIL: (2)(−5)+(2)(−i)+(−i)(−5)+(−i)(−i)=−10−2i+5i−1(2)(-5) + (2)(-i) + (-i)(-5) + (-i)(-i) = -10 - 2i + 5i - 1.

Flashcard 45: What is (−i)(i)(-i)(i) simplified?

Answer: 11. Since (−i)(i)=−i2=−(−1)=1(-i)(i) = -i^2 = -(-1) = 1.

Flashcard 46: What is (1−i)(1−i)(1-i)(1-i) in standard form?

Answer: −2i-2i. Use FOIL: (1)(1)+(1)(−i)+(−i)(1)+(−i)(−i)=1−i−i−1=−2i(1)(1) + (1)(-i) + (-i)(1) + (-i)(-i) = 1 - i - i - 1 = -2i.

Flashcard 47: What is −1(6−2i)-1(6-2i) simplified to standard form?

Answer: −6+2i-6+2i. Distribute: −1⋅6=−6-1 \cdot 6 = -6, −1⋅(−2i)=2i-1 \cdot (-2i) = 2i.

Flashcard 48: What is (a+bi)(c−di)(a+bi)(c-di) in standard form?

Answer: (ac+bd)+(bc−ad)i(ac+bd)+(bc-ad)i. Use FOIL with the second complex number's conjugate form.

Flashcard 49: What is (8+0i)−(3+0i)(8+0i)-(3+0i) in standard form?

Answer: 55. Subtract real parts only: 8−3=58-3=5, imaginary parts are zero.