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2.4 Complex Numbers

Arithmetic in the form a + bi, including dividing by the conjugate to clear i from a denominator, and simplifying high powers of i through their four-step cycle.

Algebra and Trigonometry 2e, 2.4 Complex Numbers

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41 exercises

Simplify a radical expression using the Product Property.

Simplify: ⓐ 500\sqrt{500} ⓑ 163\sqrt[3]{16} ⓒ 2434.\sqrt[4]{243} .

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2.4 Exercise 10

Simplify: ⓐ -64\sqrt{-64} ⓑ -98\sqrt{-98} ⓒ -243\sqrt{-243}

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Expressing an Imaginary Number in Standard Form

Express −9 \sqrt{−9} in standard form.

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2.4 Exercise 23

Express −24 \sqrt{−24} in standard form.

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Adding and Subtracting Complex Numbers

Add or subtract as indicated.

  1. ⓐ ( 3−4i )+( 2+5i ) \left(3 - 4 i\right) + \left(2 + 5 i\right)
  2. ⓑ ( −5+7i )−( −11+2i ) \left(−5 + 7 i\right) - \left(−11 + 2 i\right)
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2.4 Exercise 27

Subtract 2+5i 2 + 5 i from 3–4i. 3 – 4 i .

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Multiplying a Complex Number by a Real Number

Find the product 4(2+5i). 4 \left(2 + 5 i\right) .

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2.4 Exercise 29

Find the product: 1 2 ( 5−2i ). \frac{1}{2} \left(5 - 2 i\right) .

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Multiplying a Complex Number by a Complex Number

Multiply: ( 4+3i )(2−5i). \left(4 + 3 i\right) \left(2 - 5 i\right) .

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2.4 Exercise 31

Multiply: ( 3−4i )( 2+3i ). \left(3 - 4 i\right) \left(2 + 3 i\right) .

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Finding Complex Conjugates

Find the complex conjugate of each number.

  1. ⓐ 2+i 5 2 + i \sqrt{5}
  2. ⓑ − 1 2 i - \frac{1}{2} i
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2.4 Exercise 33

Find the complex conjugate of −3+4i. −3 + 4 i .

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Dividing Complex Numbers

Divide: ( 2+5i ) \left(2 + 5 i\right) by ( 4−i ). \left(4 - i\right) .

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Simplifying Powers of i

Evaluate: i 35 . i^{35} .

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2.4 Exercise 36

Evaluate: i 18 i^{18}

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2.4 Exercise 37

Explain how to add complex numbers.

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2.4 Exercise 39

Give an example to show that the product of two imaginary numbers is not always imaginary.

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2.4 Exercise 41

Evaluate the algebraic expressions.

If y= x 2 +x−4, y = x^{2} + x - 4 , evaluate y y given x=2i. x = 2 i .

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2.4 Exercise 43

Evaluate the algebraic expressions.

If y= x 2 +3x+5, y = x^{2} + 3 x + 5 , evaluate y y given x=2+i. x = 2 + i .

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2.4 Exercise 45

Evaluate the algebraic expressions.

If y= x+1 2−x , y = \frac{x + 1}{2 - x} , evaluate y y given x=5i. x = 5 i .

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2.4 Exercise 51

Perform the indicated operation and express the result as a simplified complex number.

( 3+2i )+(5−3i) \left(3 + 2 i\right) + \left(5 - 3 i\right)

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2.4 Exercise 53

Perform the indicated operation and express the result as a simplified complex number.

( −5+3i )−(6−i) \left(−5 + 3 i\right) - \left(6 - i\right)

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2.4 Exercise 55

Perform the indicated operation and express the result as a simplified complex number.

(−4+4i)−(−6+9i) \left(−4 + 4 i\right) - \left(−6 + 9 i\right)

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2.4 Exercise 57

Perform the indicated operation and express the result as a simplified complex number.

( 5−2i )(3i) \left(5 - 2 i\right) \left(3 i\right)

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2.4 Exercise 59

Perform the indicated operation and express the result as a simplified complex number.

( −2+4i )( 8 ) \left(−2 + 4 i\right) \left(8\right)

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2.4 Exercise 61

Perform the indicated operation and express the result as a simplified complex number.

( −1+2i )(−2+3i) \left(−1 + 2 i\right) \left(−2 + 3 i\right)

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2.4 Exercise 63

Perform the indicated operation and express the result as a simplified complex number.

( 3+4i )( 3−4i ) \left(3 + 4 i\right) \left(3 - 4 i\right)

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2.4 Exercise 65

Perform the indicated operation and express the result as a simplified complex number.

6−2i 3 \frac{6 - 2 i}{3}

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2.4 Exercise 67

Perform the indicated operation and express the result as a simplified complex number.

6+4i i \frac{6 + 4 i}{i}

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2.4 Exercise 69

Perform the indicated operation and express the result as a simplified complex number.

3+4i 2−i \frac{3 + 4 i}{2 - i}

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2.4 Exercise 71

Perform the indicated operation and express the result as a simplified complex number.

−9 +3 −16 \sqrt{−9} + 3 \sqrt{−16}

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2.4 Exercise 73

Perform the indicated operation and express the result as a simplified complex number.

2+ −12 2 \frac{2 + \sqrt{- 12}}{2}

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2.4 Exercise 75

Perform the indicated operation and express the result as a simplified complex number.

i 8 i^{8}

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2.4 Exercise 77

Perform the indicated operation and express the result as a simplified complex number.

i 22 i^{22}

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2.4 Exercise 79

Use a calculator to help answer the questions.

Evaluate (1−i) k \left(1 - i\right)^{k} for k=2,6, k = 2 , 6 , and 1010. Predict the value if k=14. k = 14.

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2.4 Exercise 81

Use a calculator to help answer the questions.

Show that a solution of x 6 +1=0 x^{6} + 1 = 0 is 3 2 + 1 2 i. \frac{\sqrt{3}}{2} + \frac{1}{2} i .

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2.4 Exercise 83

Evaluate the expressions, writing the result as a simplified complex number.

1 i + 4 i 3 \frac{1}{i} + \frac{4}{i^{3}}

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2.4 Exercise 85

Evaluate the expressions, writing the result as a simplified complex number.

i 7 ( 1+ i 2 ) i^{7} \left(1 + i^{2}\right)

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2.4 Exercise 87

Evaluate the expressions, writing the result as a simplified complex number.

( 2+i )( 4−2i ) (1+i) \frac{\left(2 + i\right) \left(4 - 2 i\right)}{\left(1 + i\right)}

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2.4 Exercise 89

Evaluate the expressions, writing the result as a simplified complex number.

( 3+i ) 2 ( 1+2i ) 2 \frac{\left(3 + i\right)^{2}}{\left(1 + 2 i\right)^{2}}

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2.4 Exercise 91

Evaluate the expressions, writing the result as a simplified complex number.

4+i i + 3−4i 1−i \frac{4 + i}{i} + \frac{3 - 4 i}{1 - i}

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.