Teaching Notes 7.9: Multiplying Imaginary Numbers
Multiplying imaginary numbers causes two major problems for students. The product property of
square roots does not apply if both radicands are negative, and the powers ofiare easily confused.- Review how to simplify the square roots of negative numbers by presenting this example:√
− 10 =
√
− 1 ·
√
10 =i√
- Depending on the abilities of your students, you might find it
helpful to refer to 7.8: ‘‘Simplifying Square Roots of Negative Numbers.’’ - Review the product property of square roots,
√
xy=√
x·√
y,wherexandyare real numbers
that are greater than or equal to 0. Explain that this property applies to imaginary numbers
providedxoryis negative; it does not apply if bothxandyare negative. Note that it is a
good strategy to factor out√
−1andreplaceitbyibefore applying the property.- Offer this example:
√
− 10 ·
√
−3. Explain that students must simplify each radical to
eliminate the negative numbers in both radicands before they multiply.√
− 10 ·
√
− 3 =
i√
10 ·i√
3 =i^2√
30
- Explain thati^2
√
30 is not in simplest form. Becausei=√
−1,i^2 =(√
−1)^2 =−1. Thus, in
simplest formi^2√
30 is−√
30.
- Review the information and examples on the worksheet with your students. Discuss the
examples fully, making sure your students understand all of the steps.
EXTRA HELP:
The product of two negative radicands is never a positive number. Remember to factor√
−1before
you multiply.ANSWER KEY:
(1) 6 i (2)− 10 (3)−√
6 (4)i (5)− 4 i (6)− 2 i√
2 (7)− 6 (8) 6 i√
10 (9) 8 i√
15
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(10)− 4 i√
5 (11)− 9 (12)− 20
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(Challenge)Although Vicky applied the product property of square roots to the product of two
negative radicands, the property does not apply to two negative radicands. She should have
factored√
−1 out of both radicands. The correct answer is− 2√
5.
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268 THE ALGEBRA TEACHER’S GUIDE