Teaching Notes 7.8: Simplifying Square Roots
of Negative Numbers
The concept of an imaginary number—a number that involves the imaginary uniti—is abstract.
Students often have trouble rewriting expressions by factoring out the√
−1 and simplifying
expressions usingi.- Explain that there are two types of numbers—real numbers and imaginary numbers.
Examples of real numbers include the counting numbers, zero, negative integers, rational
numbers, and irrational numbers. You might want to offer some examples, such as 3,−7,
2
5
,0.3,π,andsoon.- Explain that the square root of a negative number is an imaginary number. There is no real
number squared whose product is a negative number. Note that an imaginary number may
be simplified in a manner that is similar to simplifying radicals. You may find it helpful to
review 7.1: ‘‘Simplifying Radicals.’’ - Review the information and example on the worksheet with your students. Make sure that
your students understand the product property of square roots.
√
xy=√
x·√
ywherexand
yare real numbers that are greater than or equal to 0. Explain that when finding the square
root of a negative number, the√
−1 must be factored first, then the radical should be simpli-
fied, if possible. When numbers precede the radical, the numbers can be multiplied.EXTRA HELP:
It is customary to writeidirectly before the radical symbol. For example,i· 4√
3 should be written
as 4i√
3.
ANSWER KEY:
(1) 4 i (2) 8 i√
5 (3) 3 i√
3 (4) 2 i√
3 (5) 2 i√
30 (6) 9 i√
2 (7)i√
10 (8) 12 i√
6
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(9) 6 i√
6 (10)− 20 i√
2
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(Challenge)Marta is incorrect. She can further simplify 8i√
8as16i√
2.
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266 THE ALGEBRA TEACHER’S GUIDE