1. Chapter 7 Radicals, Radical Functions, and Rational Exponents

2. 7.1 Radical Expressions and Functions
Square Root
If a >= 0, then b, where b >= 0, such that b2 = a, is the principal square root of a
√ a = b
E.g.,
√25 = 5
√100 = 10

3.
= ----, because --- =
= 25 = 5
= = 7

4. Negative Square Root 25 = 5 ---- principal square root
Given: a
What is the square root of a?
Given: 25
What is the square root of 25?
sqrt = 5, sqrt = -5, because 52 = 25, (-5)2 = 25
Note. in - a = b, a must be >= 0

5. Your Turn Find the value of x: x = 121 x = 3 + 13 x = 36/81

6. Square Root Function f(x) = x x y = x (x, y) (0,0) 1 (1,1) 4 2 (4,2) 9
(0,0) 1
(1,1) 4 2
(4,2) 9 3
(9,3) 16
(16,4) 18
4.24
(18,4.2) x
Excel Chart

7. Evaluating a Square Root Function
Given: f(x) = 12x – 20
Find: f(3)
Solution:
f(3) = (3) – = – = = 4

8. Domain of a Square Root Function
Radicand
Given: f(x) = 3x + 12
Find the Domain of f(x):
Solution:
Radicand must be non-zero. 3x + 12 ≥ 0 3x ≥ -12 x ≥ -4 [-4, ∞)

9. Application
By 2005, an “hour-long” show on prime time TV was 45.4 min on the average, and the rest was commercials, plugs, etc. But this amount of “clutter“ was leveling off in recent years. The amount of non-program “clutter”, in minutes, was given by: M(x) = x where x is the number of years after 1996.
What was the number of minutes of “clutter” in an hour program in 2002?

10. Solution Solution: In 2009? M(x) = 0.7 x + 12.5
x = 2002 – 1996 = 6 M(6) = ~ 0.7(2.45) ~ 14.2 (min)
In 2009?
x = 2009 – 1996 = 13
M(13) = ~ 15 (min)

11. Cube Root and Cube Root Function
a = b,
means b3 = a
8 = 2,
because 23 = 8
= -4
Because (-4)3 = -64 3 3 3

12. Cube Root Function f(x) = x x y = x (x, y) -27 -3 (-27,-3) -8 -2
(-8,-2) -1
(-1,-1)
(0,0) 1
(1,1) 8 2
(8,2) 27 3
(27,3) 30
3.1
(30,3.1) 3

13. Simplifying Radical Expressions
-64x3 = (-4x)3 = -4x
81 = (3)4 = 3
-81 = x has no solution in R,
since there is no x such that x4 = -81
In general
-a has an nth root when n is odd
-a has no nth root when n is even 3 3 4 4 4 n n

14. Your Turn Simplify the following: 17 + 19 4 · 25 ±6 (-2)(2) ±10
3 125 ±6
±10
No solution 5 -3

15. 7.2 Rational Exponents What is the meaning of 71/3?
x = 71/3 means
x3 = (71/3)3 = 7
Generally, a1/n is number such that
(a1/n)n = a

16. Your Turn Simplify Solutions 641/2 (-125)1/3 (6x2y)1/3 (-8)1/3 8 -5-2 3

17. Solve 10002/3 163/2 (161/2)3 = 43 = 64 -323/5 -(321/5)3 = -(2)3 = -8
= (10001/3)2 = 102 = 100
163/2
(161/2)3 = 43 = 64
-323/5
-(321/5)3 = -(2)3 = -8

18. Negative Exponent What is the meaning of the following? 5-2
We want 5-2 · 52 = 51 = 5 Thus, 5-2 = 1/52
Examples
3-3 = 1/33 = 1/27
5-3 = 1/53 = 1/125
9-1/2 = 1/91/2 = 1/3

19. Order of Precedence What is the difference
between -323/5 and (-32) 3/ Note: 25 = 32
between -163/4 and (-16) 3/ Note: 24 = 16

20. Simplify 61/7 · 64/7 32x1/2 16x3/4 (8.33/4)2/3 = 6(1/4 + 4/7) = 65/7
= 6(1/4 + 4/7) = 65/7
32x1/2 16x3/4
= 2x(1/2 – 3/4) = 2x-1/4
(8.33/4)2/3
= 8.3(3/4 ∙ 2/3) = 8.31/2

21. Simplify 49-1/2 (8/27)-1/3 (-64)-2/3 (52/3)3 (2x1/2)5
= (72)-1/2 = 7-1 = 1/7
(8/27)-1/3
= 1/(8/27)1/3 = (27/8)1/3 = 271/3/81/3 = 3/2
(-64)-2/3
= 1/(-64)2/3 = 1/((-64)1/3)2 = 1/(-4)2 = 1/16
(52/3)3
= 52/3 ∙ 3 = 52 = 25
(2x1/2)5
25x1/2 · 5 = 32x5/2

22. 7.3 Multiplying & Simplifying Radical Expressions
Product Rule
a · b = ab or
a1/n · b1/n = (ab)1/n
Note: Factors have same order of root.
E.g,
= · 2 = = 10
2000 = · 5 = · 5 = n n n

23. Simplify Radicals by Factoring
√(80)
= √(8 · 2 · 5) = √(23 · 2 · 5) = √(24 · 5) = 4√(5)
√(40)
= √(8 · 5) = √(23 · 5) = 2√(5)
√(200x4y2)
= √(5 · 40x4y2) = √(5 · 5 · 8x4y2) = √(52 · 22 · 2x4y2) = 5 · 2x2y√(2) = 10x2y√(2)
√(80)
= √(8 · 2 · 5) = √(23 · 2 · 5) = √(24 · 5) = 4√(5)
√(40)
= √(8 · 5) = √(23 · 5) = 2√(5)
√(200x4y2)
= √(5 · 40x4y2) = √(5 · 5 · 8x4y2) = √(52 · 22 · 2x4y2) = 5 · 2x2y√(2) = 10x2y√(2) 3 3 3 3

24. Simplify Radicals by Factoring5
√(64x3y7z29)
= √(32 · 2x3y5y2z25z4) = √(25y5z25 · 2x3y2z4) = 2yz5√(2x3y2z4) 5 5 5

25. Multiplying & Simplifying
√(15)·√(3)
= √(45) = √(9·5) = 3√(5)
√(8x3y2)·√(8x5y3)
= √(64x8y5) = √(16·4x8y4y) = 2x2y√(4y) 4 4 4 4 4

26. Application
Paleontologists use the function W(x) = 4√(2x) to estimate the walking speed of a dinosaur, W(x), in feet per second, where x is the length, in feet, of the dinosaur’s leg. What is the walking speed of a dinosaur whose leg length is 6 feet?

27. W(x) = 4√(2x)
W(6) = 4√(2·6) = 4√(12) = 4√(4·3) = 8√(3) ~ 8·(1.7) ~ 14 (ft/sec) (humans: 4.4 ft/sec walking ft/sec running)

28. Your Turn Simplify the radicals √(2x/3)·√(3/2) = √((2x/3)(3/2)) = √x
√(x/3)·√(7/y)
= √((x/3)(7/y)) = √(7x/3y)
√(81x8y6)
= √(27·3x6x2y6)= 3x2y2√(3x2)
√((x+y)4)
=√((x+y)3(x+y))= (x+y)√(x+y) 4 4 4 4 3 3 3 3 3 3

29. 7.4 Adding, Subtracting, & Dividing Adding (radicals with same indices & radicands)
8√(13) + 2√(13)
= √(13) · (8 + 2) = 10√(13)
7√(7) – 6x√(7) + 12√(7)
= √(7) ·(7 – 6x + 12) = (19 – 6x)√(7)
7√(3x) - 2√(3x) + 2x2√(3x)
= √(3x)·(7 – 2 + 2x2) = (5 + 2x2) √(3x) 3 3 3 3 3 4 4 4 4 4

30. Adding 7√(18) + 5√(8) √(27x) - 8√(12x) √(xy2) + √(8x4y5)
= 7√(9·2) + 5√(4·2) = 7·3 √(2) + 5·2√(2) = 21√(2) + 10√(2) = 31√(2)
√(27x) - 8√(12x)
= √(9·3x) - 8√(4·3x) = 3√(3x) – 8·2√(3x) = √(3x)·(3 – 16) = -13√(3x)
√(xy2) + √(8x4y5)
= √(xy2) + √(8x3y3xy2) = √(xy2) + 2xy √(xy2) = √(xy2) (1 + 2xy) = (1 + 2xy) √(xy2) 3 3 3 3 3 3 3 3

31. Dividing Radical Expressions
Recall: (a/b)1/n = (a)1/n/(b)1/n
(x2/25y6)1/2
=(x2)1/2 / (25y6)1/2 =x/5y3
(45xy)1/2/(2·51/2)
= (1/2) ·(45xy/5)1/2 = (1/2) ·(9·5xy/5)1/2 = (1/2) ·3(xy)1/2 = (3/2) ·(xy)1/2
(48x7y)1/3/(6xy-2)1/3
= ((48x7y)/6xy-2))1/3 = (8x6y3)1/3 = 2x2y

32. 7.5 Rationalizing Denominators
Given: √(3) Rationalize the denominator—get rid of the radical in the denominator.
1 √(3) √(3) = √(3) √(3)

33. Denominator Containing 2 Terms
Given: √(2) + 4
Rationalize denominator
Recall: (A + B)(A – B) = A2 – B2
√(2) – (3√(2) – 4) = 3√(2) √(2) – (3√(2) )2 – (4) √(2) (3 √(2) – 4) √(2) = = –

34. Your Turn Rationalize the denominator 2 + √(5) √(6) - √(3)
2 + √(5) √(6) - √(3)
2+√(5) √(6)+√(3) √(6)+2√(3)+√(5)√(6)+√(5)√(3) = √(6) - √(3) √(6)+√(3) – √(6) + 2√(3) + √(30) +√(15) =

35. 7.6 Radical Equations Application
A basketball player’s hang time is the time in the air while shooting a basket. It is related to the vertical height of the jump by the following formula: t = √(d) / 2 A Harlem Globetrotter slam-dunked while he was in the air for 1.16 seconds. How high did he jump?

36. Solving Radical Equations
√(x) = 10
(√(x))2 = 102 x = 100
√(2x + 3) = 5
(√(2x + 3) )2 = 52 (2x + 3) = 25 2x = 22 x = 11
Check
√(2x + 3) = 5 √(2(11) + 3) = 5 ? √(22 + 3) = 5 ? √(25) = 5 ? = 5 yes

37. Solve
Check:
√(x - 3) + 6 = 5 √(4 - 3) + 6 = 5 ? √(1) = 5 ? = 5 ? False
Thus, there is no solution to this equation.
√(x - 3) + 6 = 5
√(x - 3) = -1 (√(x - 3))2 = (-1)2 (x – 3) = 1 x = 4

38. Your Turn Solve: √(x – 1) + 7 = 2
√(x – 1) = -5 (√(x – 1))2 = (-5)2 x – 1 = 25 x = 26
Check:
√(x – 1) = 2 √(26 – 1) + 7 = 2 ? √(25) = 2 ? = 2 ? False
Thus, there is no solution to this equation.

39. Your Turn Solve: x + √(26 – 11x) = 4
√(26 – 11x) = 4 – x (√(26 – 11x))2 = (4 – x)2 26 – 11x = 16 – 8x + x2 0 = x2 + 3x – 10 x2 + 3x – 10 = 0 (x – 2)(x + 5) = 0 x – 2 = 0 x = 2 x + 5 = 0 x = -5
Check -5:
√(26 – 11x) = 4 – x √(26 – 11(-5)) = 4 – (-5) ? √( ) = ? √(81) = ? = True
Check 2:
√(26 – 11x) = 4 – x √(26 – 11(2)) = 4 – 2 ? √(4) = ? = True
Solution: {-5, 2}

40. Hang Time in Basketball
A basketball player’s hang time is the time spent in the air when shooting a basket. It is a function of vertical height of jump √(d) t = where t is hang time in sec and d is vertical distance in feet.
If Michael Wilson of Harlem Globetrotters had a hang time of 1.16 sec, what was his vertical jump?

41. Hang Time
√(d) t = t = √(d) 2(1.16) = √(d) = √(d) (2.32)2 = (√(d)) = d

42. 7.7 Complex Numbers What kind of number is x = √(-25)?
Imaginary Unit i
i = √(-1), i 2 = -1
Example
√(-25) = √((25)(-1)) = √(25)√(-1) = 5i
√(-80) = √((80)(-1)) = √((16 · 5)(-1)) = 4√(5)i = 4i √(5)

43. Your Turn
Express the following with i.
√(-49)
√(-21)
√(-125)
-√(-300)

44. Complex Numbers
Comlex number has a Real part and an Imaginary part of the form: a + bi
Example
2 + 3i
-4 + 5i
5 – 2i

45. Adding and Subtracting Complex Numbers
(5 – 11i) + (7 + 4i) = 5 – 11i i = 12 – 7i
(2 + 6i) – (12 – 4i) = 2 + 6i – i = i

46. Multiplying Complex Numbers
4i(3 – 5i) = 12i – 20i2 = 12i – 20(-1) = i
(5 + 4i)(6 – 7i) = 5·6 – 5 ·7i + 4i· 6 – 4 ·7i2 = 30 – 35i + 24i – 28(-1) = 30 – 11i + 28 = 58 – 11i

47. Multiplying √(-3) √(-5) = i√(3) · i√(5) = i2 √(15) = -√(15)

48. Conjugates and Division
Given: a + bi Conjugate of a + bi: a – bi Conjugate of a – bi: a + bi
Why conjugates? (a + bi)(a – bi) = (a)2 – (bi)2 = a2 – b2i2 = a2 + b2
(3 + 2i)(3 – 2i) = 9 – (2i)2= 9 – 4(-1) = 13
Multiplying a complex number by its conjugate results in a real number.

49. Dividing Complex Numbers
Express i as a + bi – 5i
7 + 4i (7 + 4i) (2 + 5i) i + 8i = · = – 5 i (2 – 5i) (2 + 5i) – 43i =

50. Your Turn 6 + 2i -------- 4 – 3i
6 + 2i (4 + 3i) i + 8i + 6i2 = · = (4 – 3i) (4 + 3i) ( i) =

51. Your Turn
5i – i
(5i – 4) -3i i2 + 12i = · = i i i i (5 + 4i) i = = =

52. Powers of i
i2 = -1 i3 = (-1)i = -i i4 = (-1)2 = 1 i5 = (i4)i = i i6 = (-1)3 = -1 i7 = (i6)i = -i i8 = (-1)4 = 1 i9 = (i8)i = i i10 = (-1)5 = -1

53. Your Turn Simplify i17 i50 i35 i17 = i16i = (i2)8i = i

54. Application
Electrical engineers use the Ohm’s law to relate the current (I, in amperes), voltage (E, in volts), and resistence (R, in ohms) in a circuit: E = IR
Given: I = (4 – 5i) and R = (3 + 7i), what is E?
E = (4 – 5i)(3 + 7i) = i - 15i - 35i2 = i (volts)