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Algebra II : Complex Imaginary Numbers

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Example Question #11 : Complex Imaginary Numbers

Simplify:

(15-9i)-(6-10i)

Possible Answers:

1+5i

9+i

21-19i

9-i

Correct answer:

9+i

Explanation:

To subtract complex numbers, subtract the real terms, then subtract the imaginary terms.

(15-9i)-(6-10i)=(15-6)+(-9+10)i=9+i

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Example Question #2 : Irrational Numbers

Simplify:

(5+8i)(6-2i)

Possible Answers:

7+16i

16+32i

46+38i

25+8i

Correct answer:

46+38i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(5+8i)(6-2i)=30-10i+48i-16i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

30-10i+48i-16i^2=30+38i-16(-1)=30+38i+16=46+38i

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Example Question #11 : Number Theory

Simplify:

(2+5i)(-8-i)

Possible Answers:

42+11i

-11-42i

-18-5i

12+6i

Correct answer:

-11-42i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+5i)(-8-i)=-16-2i-40i-5i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

-16-2i-40i-5i^2=-16-42i-5(-1)=-16+5-42i=-11-42i

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Example Question #12 : Number Theory

Simplify:

(2+6i)(5+3i)

Possible Answers:

36-8i

-36-8i

12-8i

-8+36i

Correct answer:

-8+36i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+6i)(5+3i)=10+6i+30i+18i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

10+6i+30i+18i^2=10+36i+18(-1)=10-18+36i=-8+36i

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Example Question #13 : Number Theory

Simplify:

(2+5i)(2-5i)

Possible Answers:

16-5i

4-10i

Correct answer:

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+5i)(2-5i)=4-10i+10i-25i^2

Remember that $i=\sqrt{-1}$ , so i^2=-1 .

Substitute in for i^2 .

4-10i+10i-25i^2=4-25(-1)=4+25=29

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Example Question #1 : Complex Conjugates

Simplify:

$\frac{5-2i}{4-i}$

Possible Answers:

$\frac{5i+6}{6}$

$\frac{22-3i}{17}$

6+3i

18-9i

Correct answer:

$\frac{22-3i}{17}$

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

$\frac{5-2i}{4-i}\left(\frac{4+i}{4+i}\right)$

Now, multiply and simplify.

$\frac{5-2i}{4-i}\left(\frac{4+i}{4+i}\right)=\frac{20+5i-8i-2i^2}{16-i^2}$

Remember that i^2=-1

$\frac{20+5i-8i-2i^2}{16-i^2}=\frac{20-3i-2(-1)}{16-(-1)}=\frac{22-3i}{17}$

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Example Question #14 : Number Theory

Simplify:

$\frac{8-3i}{2-i}$

Possible Answers:

$\frac{7+2i}{3}$

3i+6

10-7i

$\frac{19+2i}{5}$

Correct answer:

$\frac{19+2i}{5}$

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

$\frac{8-3i}{2-i}\left(\frac{2+i}{2+i}\right)$

Now, multiply and simplify.

$\frac{8-3i}{2-i}\left(\frac{2+i}{2+i}\right)=\frac{16+8i-6i-3i^2}{4-i^2}$

Remember that i^2=-1

$\frac{16+8i-6i-3i^2}{4-i^2}=\frac{16+2i-3(-1)}{4-(-1)}=\frac{19+2i}{5}$

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Example Question #15 : Number Theory

Write in standard form: $\frac{3}{2+4i}$

Possible Answers:

$\frac{6-12i}{20}$

6-12i

$\frac{2+4i}{3}$

None of the other answers

$\frac{3}{10}-\frac{3i}{5}$

Correct answer:

$\frac{3}{10}-\frac{3i}{5}$

Explanation:

$\frac{3}{2+4i}$

Multiply by the conjugate:

$\frac{3}{2+4i}*\frac{2-4i}{2-4i}$

$\frac{3(2-4i)}{4-16i^2}$

$\frac{6-12i}{4-16(-1)}$

$\frac{6-12i}{20}=\frac{6}{20}-\frac{12i}{20}=\mathbf{\frac{3}{10}-\frac{3i}{5}}$

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Example Question #12 : Complex Imaginary Numbers

Write in standard form: $\frac{12i}{2+4i}$

Possible Answers:

$\frac{6i}{5}-\frac{12}{5}$

$\frac{6i}{5}-5$

$\frac{6i}{5}+\frac{12}{5}$

$\frac{6}{5}+\frac{12i}{5}$

None of these.

Correct answer:

$\frac{6i}{5}+\frac{12}{5}$

Explanation:

$\frac{12i}{2+4i}$

Multiply by the conjugate:

$\frac{12i}{2+4i}*\frac{2-4i}{2-4i}$

$\frac{12i(2-4i)}{4-16i^2}$

Combine:

$\frac{24i-48i^2}{4-16i^2}$

Simplify:

$\frac{24i+48}{20}$

$\mathbf{\frac{6i}{5}+\frac{12}{5}}$

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Example Question #4631 : Algebra Ii

Simplify:

$\sqrt{-64}$

Possible Answers:

None of the above

64i

$\pm8i$

Correct answer:

$\pm8i$

Explanation:

To solve a radical that has a negative sign under it we need to factor it first.

$\sqrt{-64} = \sqrt{(-1)*64}$

$= \sqrt{(-1)}*\sqrt{64}$

Recall that $\sqrt{-1}=i$ . Using this fact we get the following.

$= i*\pm8 = \pm8i$

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Algebra II : Complex Imaginary Numbers

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Complex Imaginary Numbers

Example Question #2 : Irrational Numbers

Example Question #11 : Number Theory

Example Question #12 : Number Theory

Example Question #13 : Number Theory

Example Question #1 : Complex Conjugates

Example Question #14 : Number Theory

Example Question #15 : Number Theory

Example Question #12 : Complex Imaginary Numbers

Example Question #4631 : Algebra Ii

All Algebra II Resources

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