Q: If $$z = 7 - 6i$$ and $$w = 3 - 4i$$, which of the following is the correct calculation for the imaginary part of $$zw$$?

$$7 \times (-4) + 3 \times (-6) = -28 - 18 = -46$$

Q: If $$z_1 = 7 - 6i$ and z_2$ = 3 - 4i$$, which of the following is the correct calculation for the real part of $$z_1$$ $$z_2$$?

$$7 \times 3 - 6 \times 4 = 21 - 24 = -3$$

Question 1

Given the complex numbers $$7 - 6i$$ and $$3 - 4i$$, what is their product?

Accepted Answer

$$21 - 28i - 18i + 24i^2$ = 21 - 46i + 24(-1) = -3 - 46i$$

Question 2

Which of the following is the correct expansion of $$(a + bi)(c + di)$$ for any complex numbers $$a, b, c, d$$?

Accepted Answer

$$(ac - bd) + (ad + bc)i$$

Question 3

If $$z_1 = 7 - 6i$ and z_2$ = 3 - 4i$$, what is the real part of $$z_1$$ $$z_2$$?

Accepted Answer

$$-3$$

Question 4

If $$z_1 = 7 - 6i$ and z_2$ = 3 - 4i$$, what is the imaginary part of $$z_1$$ $$z_2$$?

Accepted Answer

$$-46$$

Question 5

Which property of complex numbers is illustrated by the multiplication $$(7 - 6i)(3 - 4i)$$?

Accepted Answer

Distributive property

Question 6

Suppose you are calculating the product of two complex numbers in a physics problem involving alternating current. If the numbers are $$7 - 6i$$ and $$3 - 4i$$, what is the result?

Accepted Answer

$$-3 - 46i$$

Question 7

Which of the following steps is necessary when multiplying $$(7 - 6i)(3 - 4i)$$?

Accepted Answer

Replace $$i^2$$ with $$-1$$

Question 8

If $$z = 7 - 6i$$ and $$w = 3 - 4i$$, which of the following is the correct calculation for the imaginary part of $$zw$$?

Accepted Answer

$$7 	imes (-4) + 3 	imes (-6) = -28 - 18 = -46$$

Question 9

A student multiplies $$(7 - 6i)(3 - 4i)$$ and gets $$21 - 24i - 18i + 24i^2$. What mistake did the student make?

Accepted Answer

Incorrectly distributed the terms

Question 10

Which of the following is a general rule when multiplying two complex numbers $$(a + bi)(c + di)$$?

Accepted Answer

The real part is $$ac - bd$$ and the imaginary part is $$ad + bc$$

Question 11

In electrical engineering, complex numbers are used to represent impedance. If the impedance of one component is $$7 - 6i$$ ohms and another is $$3 - 4i$$ ohms, what is the combined impedance when they are multiplied?

Accepted Answer

$$-3 - 46i$$ ohms

Question 12

If you multiply $$(a + bi)$$ and $$(c + di)$$, which term in the expansion represents the product of the imaginary parts?

Accepted Answer

$$bd$$

Question 13

Which of the following is the correct simplification for $$24i^2$$?

Accepted Answer

$$-24$$

Question 14

A chemistry student is calculating the product of two complex numbers representing reactant concentrations: $$7 - 6i$$ and $$3 - 4i. $$What is the result?

Accepted Answer

$$-3 - 46i$$

Question 15

When multiplying $$(7 - 6i)(3 - 4i)$$, which step comes first?

Accepted Answer

Apply the distributive property (FOIL method)

Question 16

If $$z_1 = 7 - 6i$ and z_2$ = 3 - 4i$$, which of the following is the correct calculation for the real part of $$z_1$$ $$z_2$$?

Accepted Answer

$$7 	imes 3 - 6 	imes 4 = 21 - 24 = -3$$

Question 17

Which of the following is NOT a correct step in multiplying $$(7 - 6i)(3 - 4i)$$?

Accepted Answer

Add the coefficients of $$i$$ directly

Question 18

If you multiply $$(a + bi)$$ and $$(c + di)$$, what is the coefficient of $$i in $$the result?

Accepted Answer

$$ad + bc$$

Question 19

A student claims that $$(7 - 6i)(3 - 4i) = 21 - 24i - 18i + 24i^2$. What is the correct simplification of this expression?

Accepted Answer

$$-3 - 46i$$

Complex Numbers: Multiplication and Applications

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