Operations with Complex Numbers - Algebra 2

1. What is $(10-3i)-(4+9i)$ in standard form?

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

2. What is $(2i)(5-4i)$ in standard form?

   $8+10i$. Distribute: $(2i)(5) + (2i)(-4i) = 10i + 8i^2 = 10i - 8$.

3. What is $(5-2i)(1+3i)$ in standard form?

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

4. What is $(7i)(-3i)$ simplified?

   $21$. Since $(7i)(-3i) = -21i^2 = -21(-1) = 21$.

5. What is the imaginary part of the complex number $a+bi$?

   $b$. The coefficient of $i$ in standard form.

6. What is the value of $i^5$ written in simplest form?

   $i$. Since $i^5 = i^4 \cdot i = 1 \cdot i = i$.

7. What is the value of $i^2$?

   $-1$. By definition of the imaginary unit.

8. What is $(2+i)(3+4i)$ in standard form?

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

9. What is $(-5i)+(12i)$ in standard form?

   $7i$. Add imaginary parts: $-5i + 12i = 7i$.

10. What is the multiplicative identity for complex numbers?

    $1+0i$. Multiplying any complex number by this yields the same number.

11. What is $(3+2i)+(5-7i)$ in standard form?

    $8-5i$. Add real parts: $3+5=8$, imaginary parts: $2+(-7)=-5$.

12. What is $(1+i)(1-i)$ in standard form?

    $2$. Difference of squares: $(1)^2 - (i)^2 = 1 - i^2 = 1 + 1 = 2$.

13. What is $(-3+4i)-(8+i)$ in standard form?

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

14. What is the standard form of a complex number?

    $a+bi$. Where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

15. What is $(a+bi)(a-bi)$ simplified using $i^2=-1$?

    $a^2+b^2$. Conjugate multiplication yields sum of squares.

16. What is $(4-9i)+(-6+3i)$ in standard form?

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

17. What is $(3-2i)(-1+i)$ in standard form?

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

18. What is $(2-3i)^2$ in standard form?

    $-5-12i$. Square using $(a-b)^2 = a^2 - 2ab + b^2$ and $i^2 = -1$.

19. What is $(2+5i)(-3-2i)$ in standard form?

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

20. What is $(3+i)^2$ in standard form?

    $8+6i$. Square using $(a+b)^2 = a^2 + 2ab + b^2$ and $i^2 = -1$.

21. What is $(bi)(di)$ simplified using $i^2=-1$?

    $-bd$. Since $(bi)(di) = bdi^2 = bd(-1) = -bd$.

22. What is $(2+3i)+(2-3i)$ in standard form?

    $4$. The imaginary parts cancel: $2+2=4$, $3i+(-3i)=0$.

23. What is the value of $i^4$ written in simplest form?

    $1$. Since $i^4 = (i^2)^2 = (-1)^2 = 1$.

24. What is $2(3-5i)$ simplified to standard form?

    $6-10i$. Distribute: $2 \cdot 3 = 6$, $2 \cdot (-5i) = -10i$.

25. What is $(4+3i)(4-3i)$ in standard form?

    $25$. Difference of squares: $(4)^2 - (3i)^2 = 16 - 9i^2 = 16 + 9 = 25$.

26. What is $(1-2i)(-4+3i)$ in standard form?

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

27. What is $(3+4i)(-3+4i)$ in standard form?

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

28. What is the additive identity for complex numbers?

    $0+0i$. Adding this to any complex number yields the same number.

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

    $(ac-bd)+(ad+bc)i$. Use FOIL and $i^2 = -1$ to simplify.

30. What is $(6+i)(-2+3i)$ in standard form?

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

31. What is $(1+i)(1+i)$ in standard form?

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

32. What is $(a+bi)+(c-di)$ simplified in standard form?

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

33. What is the difference $(a+bi)-(c+di)$ in standard form?

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

34. What is $-4(-2+3i)$ simplified to standard form?

    $8-12i$. Distribute: $-4 \cdot (-2) = 8$, $-4 \cdot 3i = -12i$.

35. What is $(7+i)-(2-5i)$ in standard form?

    $5+6i$. Subtract real parts: $7-2=5$, imaginary parts: $1-(-5)=6$.

36. What is the value of $i^3$ written in simplest form?

    $-i$. Since $i^3 = i^2 \cdot i = (-1) \cdot i = -i$.

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

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

38. What is $(1+6i)(1-6i)$ in standard form?

    $37$. Difference of squares: $(1)^2 - (6i)^2 = 1 - 36i^2 = 1 + 36 = 37$.

39. What is $(2+ i)(2- i)$ in standard form?

    $5$. Difference of squares: $(2)^2 - (i)^2 = 4 - i^2 = 4 + 1 = 5$.

40. What is the value of $i^6$ written in simplest form?

    $-1$. Since $i^6 = (i^2)^3 = (-1)^3 = -1$.

41. What is $(a+bi)-(c-di)$ simplified in standard form?

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

42. What is $(9- i)+( -9+ i)$ in standard form?

    $0$. The real and imaginary parts are additive inverses.

43. What is the real part of the complex number $a+bi$?

    $a$. The coefficient of the non-imaginary term.

44. What is $(2- i)(-5- i)$ in standard form?

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

45. What is $(-i)(i)$ simplified?

    $1$. Since $(-i)(i) = -i^2 = -(-1) = 1$.

46. What is $(1-i)(1-i)$ in standard form?

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

47. What is $-1(6-2i)$ simplified to standard form?

    $-6+2i$. Distribute: $-1 \cdot 6 = -6$, $-1 \cdot (-2i) = 2i$.

48. What is $(a+bi)(c-di)$ in standard form?

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

49. What is $(8+0i)-(3+0i)$ in standard form?

    $5$. Subtract real parts only: $8-3=5$, imaginary parts are zero.

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