Download 2.4 Tools for Operating With Complex Numbers

2.4 Tools for Operating
With Complex
Numbers
Solar cells are attached to the surfaces of
satellites. The cells convert the energy of
sunlight to electrical energy. Solar cells are
made in various shapes to cover most of the
surface area of satellites.
I NVESTIGATE & I NQUIRE
The scale drawing shows 6 solar cells. The
3 triangles and 3 rectangles are attached
to form one triangular solar panel. The
dimensions shown are in centimetres.
2
E
3
4
4
2
2
2
A
4
3
3
3
4
B
C
D
Calculate the lengths of AB, BC, and
CD. Write your answers as mixed radicals in simplest form.
1.
Explain why the three mixed radicals in simplest form from question 1
are called like radicals.
2.
Use the large right triangle ADE to write an expression for the length
of AD. Write your answer as a mixed radical in simplest form.
3.
4.
How is the length of AD related to the lengths of AB, BC, and CD?
Compare the radical expressions you wrote for the lengths of AB, BC,
CD, and AD. Then, write a rule for adding like radicals.
5.
2.4 Tools for Operating With Complex Numbers • MHR 135
6. Technology
a)
32 + 42
Use a calculator to test your rule for each of the following.
b) 7 + 2
7 + 3
7
Simplify.
a) 3
5 + 6
5
7.
b)
43 + 53 + 3
Describe a method for using information from the same diagram to
write a rule for subtracting like radicals.
b) Write and test the rule.
c) Simplify 5
6 − 2
6.
8. a)
EXAMPLE 1 Adding and Subtracting Radicals
Simplify.
a)
12
+ 18
− 27
+ 8
b)
43 + 320
− 12
+ 645
SOLUTION
Simplify radicals and combine like radicals.
a)
b)
12
+ 18
− 27
+ 8 = 4 × 3 + 9 × 2 − 9 × 3 + 4 × 2
= 23 + 32 − 33 + 22
= −3 + 52
43 + 320
− 12
+ 645
= 43 + 3 × 4 × 5 − 4 × 3 + 6 × 9 × 5
= 43 + 3 × 25 − 23 + 6 × 35
= 43 + 65 − 23 + 185
= 23 + 245
EXAMPLE 2 Multiplying a Radical by a Binomial
Expand and simplify 32 (26 + 10
).
SOLUTION
Use the distributive property.
) = 32(26 + 10
)
32(26 + 10
= 3
2 × 2
6 + 3
2 × 10
= 6
12 + 3
20
= 6 × 2
3 + 3 × 2
5
= 12
3 + 6
5
136 MHR • Chapter 2
EXAMPLE 3 Binomial Multiplication
Simplify (32 + 45 )(42 − 35 ).
SOLUTION
Multiply each term in the first binomial by each term in the second
binomial.
F
O
(32 + 45 )(42 − 35 ) = (32 + 45 )(42 − 35 )
Recall that FOIL means First, Outside, Inside, Last.
I
L
= 124 − 910
+ 1610
− 1225
= 24 − 910
+ 1610
− 60
10
= −36 + 7
Recall that a radical is in simplest form when no radical appears in the
denominator of a fraction.
EXAMPLE 4 Fractions With Radicals in the Denominator
1
Simplify .
3
2
SOLUTION
Multiply the numerator and denominator by 2.
This is the same as multiplying the fraction by 1.
2
1
1
=×
32
32
2
1 × 2
=
32 × 2
2
=
3×2
2
=
6
2.4 Tools for Operating With Complex Numbers • MHR 137
The process shown in Example 4 is called rationalizing the denominator. The
denominator has been changed from an irrational number to a rational number.
Binomials of the form ab + cd and ab − cd, where a, b, c, and d are rational
numbers, are conjugates of each other. The product of conjugates is always a
rational number.
EXAMPLE 5 Multiplying Conjugate Binomials
Simplify (7 + 23 )(7 − 23 ).
SOLUTION
F
O
(7 + 23 )(7 − 23 ) = (7 + 23 )(7 − 23 )
I
L
= 49
− 221
+ 221
− 49
= 7 − 12
= −5
Conjugate binomials can be used to simplify a fraction with a binomial radical
in the denominator.
EXAMPLE 6 Rationalizing Binomial Denominators
5
Simplify .
26 − 3
SOLUTION
Multiply the numerator and the denominator by the conjugate of 26 − 3,
which is 26 + 3.
5
5
26 + 3
=×
26 − 3
26 − 3 26 + 3
5(26 + 3 )
= 436
− 9
106 + 53
= 24 − 3
106 + 53
= 21
138 MHR • Chapter 2
Key
Concepts
• To simplify radical expressions, express radicals in simplest radical form and
add or subtract like radicals.
• To multiply binomial radical expressions, use the distributive property and add
or subtract like radicals.
• To simplify a radical expression with a monomial radical in the denominator,
multiply the numerator and the denominator by this monomial radical.
• To simplify a radical expression with a binomial radical in the denominator,
multiply the numerator and the denominator by the conjugate of the
denominator.
Communicate
Yo u r
Understanding
Explain the meaning of the term like radicals.
Describe how you would simplify each of the following.
a) 24
+ 54
b) 22
(10
− 32)
5
2
c) (3
+ 5
)(23 − 45
)
d) e) 6
5 + 3
1.
2.
Practise
A
1. Simplify.
a) 25
+ 35
+ 65
b) 43
+ 23
− 3
c) 62
− 2 + 72 − 32
d) 57
+ 37
− 27
e) 810
− 210
− 710
f) 2
− 32 − 92 + 112
g) 5
+ 5
+ 5
+ 5
Simplify.
a) 53
+ 26 + 33
b) 85
− 37 + 77 − 45
c) 22
+ 310
+ 52 − 410
2.
76 − 413
− 13
+ 6
e) 911
− 11
+ 614
− 314
− 211
f) 127
+ 9 − 37 + 4
g) 8 + 711
− 9 − 911
d)
Simplify.
a) 12
+ 27
b) 20
+ 45
c) 18
− 8
d) 50
+ 98
− 2
e) 75
+ 48
+ 27
f) 54
+ 24
+ 72
− 32
g) 28
− 27
+ 63
+ 300
3.
2.4 Tools for Operating With Complex Numbers • MHR 139
Simplify.
a) 87
+ 228
b) 350
− 232
c) 527
+ 448
d) 38
+ 18
+ 32
e) 5
+ 245
− 320
f) 43
+ 3
20 − 212
+ 45
g) 348
− 48 + 427
− 272
4.
Expand and simplify.
a) 2
(10
+ 4)
b) 3
(6
− 1)
c) 6
(2 + 6
)
d) 22
(36
− 3 )
e) 2
(3 + 4)
f) 32
(26 + 10
)
g) (5
+ 6
)(5 + 36 )
h) (23
− 1)(33 + 2)
i) (47
− 32 )(27 + 52 )
j) (33
+ 1)2
k) (22
− 5
)2
l) (2 + 3
)(2 − 3 )
m) (6
− 2
)(6 + 2
)
n) (27
+ 35 )(27
− 35
)
5.
Simplify.
1
a) 3
6.
b)
2
5
c)
2
7
d)
1
2
e)
55
23
f)
22
18
g)
42
8
h)
35
3
i)
47
214
j)
36
410
k)
711
23
l)
25
52
Simplify.
1
a) 2 + 2
7.
b)
3
5 − 1
c)
2
6 − 3
d)
2
6
+ 3
e)
3
5
− 2
f)
3
3 + 2
g)
26
26 + 1
h)
2 − 1
2 + 1
i)
2 + 5
6 − 10
j)
27
+ 5
37
− 25
Apply, Solve, Communicate
Express the perimeter of the quadrilateral in
simplest radical form.
8. Measurement
80
45
B
9. Without using a calculator, arrange the following expressions in order
from greatest to least.
3(3 + 1), (3 + 1)(3 − 1), (1 − 3 )2, (3
+ 1)2
140 MHR • Chapter 2
5
20
Without using a calculator, decide which of the following radical
expressions does not equal any of the others.
60
4
8
4
+
− 42
68
+ 8 − 58
62
450
2
18
18
b) Communication How is the radical expression you identified in part a)
related to each of the others?
10. a)
11. Nature Many aspects of nature, including the number of pairs of rabbits
in a family and the number of branches on a tree, can be described using the
Fibonacci sequence. This sequence is 1, 1, 2, 3, 5, 8, …
The expression for the n th term of the Fibonacci sequence is called Binet’s
formula. The formula is
n 1 1 − 5 n
1 1 + 5
Fn = − .
2
2
5
5
Use Binet’s formula to find F2.
12. Measurement Write and simplify
a) the area of the rectangle
b) the perimeter of the rectangle
an expression for
4 2–
3
2 3
Write and simplify an expression for
the area of the square.
13. Measurement
8–
5
Express the volume of the
rectangular prism in simplest radical form.
14. Measurement
5 2–2 3
5 2+2 3
15 – 2
If a rectangle has an area of 4 square units and a width of
7
− 5 units, what is its length, in simplest radical form?
15. Application
Write a quadratic equation in the form
ax + bx + c = 0 with the given roots.
a) 3 + 2
and 3 − 2
b) −1 + 23
and −1 − 23
13
13
c) 1 + and 1 − 2
2
16. Inquiry/Problem Solving
2
2.4 Tools for Operating With Complex Numbers • MHR 141
C
Simplify.
3
a) 16
+ 54
3
3
c) 2(32
) + 5(108
)
3
3
e) 16
− 54
3
3
g) 2(40
) − 5
17.
3
b)
3
3
24
+ 81
54
+ 5(16
)
3
3
f) 108
− 32
3
3
h) 5(48
) − 2(162
)
d)
3
3
Express the ratio of
the area of the larger circle to the area of
the smaller circle in simplest radical form.
18. Measurement
2+ 3
2–
3
State the perimeter of each of the following
triangles in simplest radical form.
19. Coordinate geometry
a)
6 A
4
2
3 D
E
C
–2
6
4
B
–4
y
b)
y
0
2
4
x
– 5
F
0
2
x
Is the statement a
+ b = a + b always true, sometimes
true, or never true? Explain.
20. Equation
LOGIC
Power
Suppose intercity buses travel from Montréal to Toronto and from Toronto to
Montréal, leaving each city on the hour every hour from 06:00 to 20:00. Each
trip takes 5 1 h. All buses travel at the same speed on the same highways. Your
2
driver waves at each of her colleagues she sees driving an intercity bus in the
opposite direction. How many times would she wave during the journey if your
bus left Toronto at
a) 14:00?
b) 18:00?
c) 06:00?
142 MHR • Chapter 2