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
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
