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Chapter 7 Radicals, Radical Functions, and Rational Exponents

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7.1 Radical Expressions and Functions

- Square Root
- If a>= 0, then b >= 0, such that b2 = a, is the principal square root of a
- √ a = b
- E.g.,
- √25 = 5
- √100 = 10

Negative Square Root

- 25 = 5 ---- principal square root
- - 25 = -5 ---- negative 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

Evaluating a Square Root Function

- Given: f(x) = 12x – 20
- Find: f(3)
- Solution:
- f(3) = 12(3) – 20 = 36 – 20 = 16 = 4

Domain of a Square Root Function

- Given: f(x) = 3x + 12
- Find the Domain of f(x):
- Solution:
- 3x + 12 ≥ 03x ≥ -12X ≥ -4[-4, ∞)

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) = 0.7 x + 12.5where x is the number of years after 1996.
- What was the number of minutes of “clutter” in an hour program in 2002?

Solution

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

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

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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

7.2 Rational Exponents

- What is the meaning of 71/3?
- x = 71/3means
- X3 = (71/3)3 = 7
- Generally, a1/n is number such that
- (a1/n)n = a

Solve

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

Your Turn

- What is the difference
- between -323/5 and (-32) 3/5
- between -163/4 and (-16) 3/4

Simplify

- 61/7· 64/7
- = 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

Simplify

- 49-1/2
- = (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

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7.3 Multiplying & SimplifyingRadical Expressions- Product Rule
- a · b = ab or
- a1/n· b1/n = (ab)1/n
- Note: Factors have same order of root.
- E.g,
- 25 4 = 25 · 4 = 100 = 10
- 2000 = 400 · 5 = 400 · 5 = 20 5

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)

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Simplify Radicals by Factoring

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- √(64x3y7z29)
- = √(32 · 2x3y5y2z25z4)= √(25y5z25· 2x3y2z4)= 2yz5√(2x3y2z4)

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Multiplying & Simplifying

- √(15)·√(3)
- = √(45) = √(9·5) = 3√(5)
- √(8x3y2)·√(8x5y3)
- = √(64x8y5) = √(16·4x8y4y)= 2x2y√(4y)

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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?

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 22 ft/sec running)

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)

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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)

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Adding

- 7√(18) + 5√(8)
- = 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)

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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

7.5 Rationalizing Denominators

- Given: 1 √(3)Rationalize the denominator—get rid of the radical in the denominator.
- 1 √(3) √(3) = √(3) √(3) 3

Denominator Containing 2 Terms

- Given: 8 3√(2) + 4
- Rationalize denominator
- Recall: (A + B)(A – B) = A2 – B2
- 8 3√(2) – 4 8(3√(2) – 4) =3√(2) + 4 3√(2) – 4 (3√(2) )2 – (4)224 √(2) - 32 8(3 √(2) – 4) 12 √(2) - 16 = = 18 – 16 2

Your Turn

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

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) / 2A Harlem Globetrotter slam-dunked while he was in the air for 1.16 seconds. How high did he jump?

Solving Radical Equations

- √(x) = 10
- (√(x))2 = 102x = 100
- √(2x + 3) = 5
- (√(2x + 3) )2 = 52(2x + 3) = 252x = 22x = 11

Check

√(2x + 3) = 5 √(2(11) + 3) = 5 ? √(22 + 3) = 5 ?√(25) = 5 ? 5 = 5 yes

Solve

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

- √(x - 3) + 6 = 5
- √(x - 3) = -1(√(x - 3))2 = (-1)2(x – 3) = 1x = 4

Your Turn

- Solve: √(x – 1) + 7 = 2
- √(x – 1) = -5(√(x – 1))2 = (-5)2x – 1 = 25x = 26

Check:

√(x – 1) + 7 = 2√(26 – 1) + 7 = 2 ?√(25) + 7 = 2 ?5 + 7 = 2 ? False

Thus, there is no solution to this equation.

Your Turn

Check -5:

√(26 – 11x) = 4 – x√(26 – 11(-5)) = 4 – (-5) ?√(26 + 55) = 4 + 5 ?√(81) = 9 ?9 = 9 True

Check 2:

√(26 – 11x) = 4 – x√(26 – 11(2)) = 4 – 2 ?√(4) = 2 ?2 = 2 True

Solution: {-5, 2}

- Solve: x + √(26 – 11x) = 4
- √(26 – 11x) = 4 – x(√(26 – 11x))2 = (4 – x)226 – 11x = 16 – 8x + x20 = x2 + 3x – 10x2 + 3x – 10 = 0(x – 2)(x + 5) = 0x – 2 = 0x = 2x + 5 = 0x = -5

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 2 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?

Hang Time

- √(d)t = ----- 2 2t = √(d)2(1.16) = √(d)2.32 = √(d)(2.32)2 = (√(d))25.38 = d

7.7 Complex Numbers

- What kind of number is x = √(-25)?
- x2 = -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)

Your Turn

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

Complex Numbers

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

Adding and Subtracting Complex Numbers

- (5 – 11i) + (7 + 4i)= 5 – 11i + 7 + 4i= 12 – 7i
- (2 + 6i) – (12 – 4i)= 2 + 6i – 12 + 4i= -10 + 10i

Multiplying Complex Numbers

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

Multiplying

- √(-3) √(-5)= i√(3) · i√(5)= i2 √(15)= -√(15)
- √(-5) √(-10)= i√(5) · i√(10)= i2 √(50)= -√(50)= -√(25 · 2)= -5√(2)

Conjugates and Division

- Given: a + biConjugate of a + bi: a – biConjugate 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.

Dividing Complex Numbers

- Express 7 + 4i -------- as a + bi 2 – 5i
- 7 + 4i (7 + 4i) (2 + 5i) 14 + 35i + 8i + 20-------- = ---------- · ----------- = ------------------------2 – 5 i (2 – 5i) (2 + 5i) 4 + 25 34 – 43i= ------------- 29

Your Turn

- 6 + 2i--------4 – 3i
- 6 + 2i (4 + 3i) 24 + 18i + 8i + 6i2= ---------- · ---------- = ------------------------- (4 – 3i) (4 + 3i) 16 + 9 (18 + 26i)= ------------- 25

Your Turn

- 5i – 4------- 3i
- (5i – 4) -3i -15i2 + 12i= --------- · ----- = ------------------- 3i -3i -9i2 15 + 12i 3(5 + 4i) 5 + 4i= ------------ = ----------- = --------- 9 9 3

Powers of i

- i2 = -1i3 = (-1)i = -ii4 = (-1)2 = 1i5 = (i4)i = ii6 = (-1)3 = -1i7 = (i6)i = -ii8 = (-1)4 = 1i9 = (i8)i = ii10 = (-1)5 = -1

Your Turn

- Simplify
- i17
- i17 = i16i = (i2)8i = i
- i50
- i50 = (i2)25 = (-1)25 = -1
- i35
- i35 = (i34)i = (i2)17i = (-1)17i = -i

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) = 12 + 28i - 15i - 35i2 = 47 + 13i (volts)

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