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Algebra II : Basic Operations with Complex Numbers

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Example Question #1 : Complex Numbers

Divide: $\frac{2+3i}{i}$

Answer must be in standard form.

Possible Answers:

3-2i

3+2i

2i-3

Correct answer:

3-2i

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator which is resulting in

$\frac{\left ( 2+3i \right )\left ( -i \right )}{\left ( i \right )\left ( -i \right )}$

This is equal to $\frac{-2i-{3i}^2}{-i^2}$

Since i^2 = -1 you can make that substitution of in place of i^2 in both numerator and denominator, leaving:

$\frac{-2i-3(-1)}{-(-1)}$

When you then cancel the negatives in both numerator and denominator (remember that -(-1)=1 , simplifying each term), you're left with a denominator of and a numerator of -2i + 3 , which equals 3-2i .

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Example Question #1 : Complex Numbers

Evaluate: i(1+i)(2-i)

Possible Answers:

1-3i

1+3 i

-1-3i

-3+i

-1+3i

Correct answer:

-1+3i

Explanation:

Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

i(1+i)(2-i)

=i[(1)(2)+(1)(-i)+(i)(2)+(i)(-i)]

=i[2-i+2i-i^2]

=i[2+i-i^2]

=2i+i^2-i^3

The imaginary is equal to:

$\sqrt{-1}$

Write the terms for $i, i^2, \textup{and } i^3$ .

$i=\sqrt{-1}$

i^2=-1

$i^3=i \times i^2=-\sqrt{-1}=-i$

Replace $i^2 \textup{ and } i^3$ with the appropiate values and simplify.

2i+i^2-i^3= 2i-1-(-i) = -1+3i

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Example Question #4691 : Algebra Ii

What is the value of $i^{42}$ , if $\small i$ = $\small \sqrt{-1}$ ?

Possible Answers:

$\small -i$

$\small i$

$\textup{No solution}$

Correct answer:

Explanation:

We know that $\small i=\sqrt{-1}}$ . Therefore, $\small i^{2}=-1, i^{3}=-\sqrt{-1}, i^{4}=1$ . Thus, every exponent of $\small i$ that is a multiple of 4 will yield the value of $\small 1$ . This makes $\small i^{40} =1$ . Since $\small i^{42}=i^{40}*i*i$ , we know that $\small i^{42}=1*i*i=i^{2}=-1$ .

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Example Question #3 : Complex Numbers

(3+2i)+4-i-i(7+i)

Possible Answers:

7(1-i)

8-6i

7-6i

The answer is not present.

Correct answer:

8-6i

Explanation:

(3+2i)+4-i-i(7+i)

Combine like terms:

7+i-i(7+i)

Distribute:

7+i-7i-i^2

Combine like terms:

7+i-7i-(-1)

$\mathbf{8-6i}$

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Example Question #4692 : Algebra Ii

$\sqrt{-4}*\sqrt{-25}*\sqrt{-64}$

Possible Answers:

80i^3

80i

-8i

-80i

Correct answer:

-80i

Explanation:

$\sqrt{-4}*\sqrt{-25}*\sqrt{-64}$

2i*5i*8i=80i^3

i^3=-1

$\mathbf{-80i}$

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Example Question #4 : Complex Numbers

Rationalize the complex fraction: $\frac{2+i}{3-i}$

Possible Answers:

$\frac{1+i}{2}$

$\frac{7+5i}{10}$

$\frac{5+i}{10}$

$\frac{5+5i}{8}$

$\frac{1-i}{2}$

Correct answer:

$\frac{1+i}{2}$

Explanation:

To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.

$\frac{(2+i)(3+i)}{(3-i)(3+i)}$

$\frac{6+3i+2i+i^2}{9-3i+3i-i^2}$

$\frac{5+5i}{10}$

$\frac{1+i}{2}$

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Example Question #4 : Complex Numbers

Rationalize the complex fraction: $\frac{2+i}{3-i}$

Possible Answers:

$\frac{1+i}{2}$

$\frac{7+5i}{10}$

$\frac{5+i}{10}$

$\frac{5+5i}{8}$

$\frac{1-i}{2}$

Correct answer:

$\frac{1+i}{2}$

Explanation:

To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.

$\frac{(2+i)(3+i)}{(3-i)(3+i)}$

$\frac{6+3i+2i+i^2}{9-3i+3i-i^2}$

$\frac{5+5i}{10}$

$\frac{1+i}{2}$

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Example Question #11 : Basic Operations With Complex Numbers

Multiply: (3i + 1 )(3 - 4i)

Possible Answers:

3-7i

2 - 12i

5i-9

5i+15

Correct answer:

5i+15

Explanation:

Distribute:

3i(3-4i) = 9i + 12

1(3-4i) = 3 - 4i

combine like terms:

5i + 15

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Example Question #5 : Complex Numbers

Multiply: (3+i)(5-4i)

Possible Answers:

11-7i

15-11i

19-7i

15-3i

19+17i

Correct answer:

19-7i

Explanation:

Use FOIL to multiply the two binomials.

Recall that FOIL stands for Firsts, Outers, Inners, and Lasts.

(3+i)(5-4i)

15+5i-12i-4i^2

Remember that i^2=-1

15-7i+4

19-7i

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Example Question #12 : Basic Operations With Complex Numbers

Simplify: (5-3i)-(2+2i)

Possible Answers:

3-i

3-5i

7+i

7-i

Correct answer:

3-5i

Explanation:

Distribute the minus sign:

5-3i-2-2i

Combine like terms:

3-5i

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Algebra II : Basic Operations with Complex Numbers

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Complex Numbers

Example Question #1 : Complex Numbers

Example Question #4691 : Algebra Ii

Example Question #3 : Complex Numbers

Example Question #4692 : Algebra Ii

Example Question #4 : Complex Numbers

Example Question #4 : Complex Numbers

Example Question #11 : Basic Operations With Complex Numbers

Example Question #5 : Complex Numbers

Example Question #12 : Basic Operations With Complex Numbers

All Algebra II Resources

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