We can use this definition to define any square root of a negative real number.
SECTION 8.7 Complex Numbers 475
For any positive real number b, 2 bi 2 b.
2 bSimplifying Square Roots of Negative NumbersWrite each number as a product of a real number and i.
(a) (b)
(c) (d)
NOW TRY2 - 2 =i 22 2 - 54 =i 254 =i 29 # 6 = 3 i 26
2 - 100 =i 2100 = 10 i - 2 - 36 = -i 236 = - 6 i
NOW TRY EXAMPLE 1
EXERCISE 1
Write each number as a prod-
uct of a real number and i.
(a) (b)
(c) 2 - 3 (d) 2 - 32
2 - 49 - 2 - 121
CAUTION It is easy to mistake for with the iunder the radical. For
this reason, we usually write 22 ias i 22 ,as in the definition of 2 - b.
22 i 22 i,
When finding a product such as we cannot use the product rule
for radicals because it applies only to nonnegativeradicands. For this reason, we
change to the form before performing any multiplications or divisions.
Multiplying Square Roots of Negative NumbersMultiply.
(a)
Take square roots.
Multiply.
Substitute for=- 6
= 61 - 12 - 1 i^2.
= 6 i^2
=i# 2 #i# 3
=i 24 #i 29 2 - b=i 2 b
2 - 4 # 2 - 9
EXAMPLE 2
2 b i 2 b
2 - 4 # 2 - 9 ,
First write all
square roots in
terms of i.(b)
Product rule=- 221
= 1 - 12221 Substitute -1 for i^2.
=i^223 # 7
=i 23 #i 27 2 - b=i 2 b
2 - 3 # 2 - 7
(c)
Product rule=- 4 Take the square root.
= 1 - 12216 i^2 =- 1
=i^222 # 8
=i 22 #i 28 2 - b=i 2 b
- 2 #- 8
First write all
square roots in
terms of i.(d)
=i 230
=i 25 # 26
2 - 5 # 26
NOW TRYNOW TRY
EXERCISE 2
Multiply.
(a)
(b)
(c)
(d) 213 # 2 - 2
2 - 3 # 2 - 12
2 - 5 # 2 - 11
2 - 4 # 2 - 16
NOW TRY ANSWERS
- (a) 7 i (b)
(c) (d)- (a) (b)
(c) - 6 (d)i 226- 8 - 255
i 23 4 i 22- 11 i