π π₯ β ,β / , 0 - los rios community college...
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FLC Ch 7
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Math 120 Intermediate Algebra Sec 7.1: Radical Expressions and Functions
Ex 1 For each number, find all of its square roots.
4 121 25
64
Ex 2 Simplify.
β1
β49
ββ81
β625
β83
ββ273
ββ13
β164
ββ164
β130
ββ180
5
ββ225
ββ225
ββ1
32
5
Ex 3 Simplify. (Assume all variables represent any real number.) AAVRARN
β16π‘2 β(π + 3)2 βπ¦33 βπ¦88
β(7π)44 β7π99
β(β4)66 β(β4)77
β9 β 6π¦ + π¦2
Ex 4 Simplify. Assume that no radicands were formed by raising negative quantities to even powers. (Assume that variables represent any positive real number.) AAVR+N
β25π‘2 ββ(7π₯π¦)2 ββ(7π₯π¦)4
ββ64π¦63 βπ14 β(π₯ + 3)10
βπππππππππππππ₯
β = β2
, β3
, β4
, etc.
βππ’π π‘ ππ β₯ 0ππ£ππ
βπππ¦ ππππ ππ’πππππππ
β( )ππ= {
| |, ππ π ππ ππ£ππ( ), ππ π ππ πππ
FLC Ch 7
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Ex 5 Find the domain of each function.
Evaluate: π(0), π(1), and π(β2)
Sec 7.2: Rational Numbers and Exponents
What does ππ/π mean? ππ/π?
Examples:
91/2 = (32)1/2 = 31 = 3 81/3
161/2 271/3
491/2 1251/3
Recall: Exponential Rules
π₯π β π₯π = π₯π+π π₯π
π₯π = π₯πβπ , π₯ β 0 (π₯π)π = π₯ππ
(π₯π¦)π = π₯ππ¦π (π₯
π¦)
π=
π₯π
π¦π π₯βπ =1
π₯π , π₯ β 0
π₯0 = 1, π₯ β 0
Conclusion: ( )1/2 = β and ( )1/3 = β3
In general, ππ/π = βππ . Note: If π is even, π must be β₯ π. If π is odd, π can be anything.
ππ/π = βπππ= ( βπ
π)
π if βπ
π exists.
π(π₯) = β4 + 3π₯
π(π₯) = β4 + 3π₯3
β(π₯) = β4 + 3π₯6
π (π₯) = 13π₯ β β4 + 3π₯
FLC Ch 7
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Ex 6 Write an equivalent expression using radical/exponential notation, and if possible, simplify. Which numbers are rational? Irrational?
π‘1/4 (π2π)1/5 47/2 (9π¦6)3/2
β104
β22 βπ45 βππ
3
Ex 7 Rewrite but do not simplify. (β2π₯π¦2π§3
)5 βπ₯2π¦5π§73
Ex 8 Simplify. Do not use negative exponents in answer. Which numbers are rational? Irrational?
51/4 β 51/8 87/11
8β2/11 (55/4)
3/7 (274/9π₯β1/3π¦2/5)
3/2
Ex 9 Simplify. Present answers in radical form.
βπ312 (βππ
3)
15 β(3π₯)28
βππ153
ββπ₯6
(βπ₯2π¦53)
12 ββ2π
35
Several problems from βWriting Expressions as Powers of βxββ handout page 38.
FLC Ch 7
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Sec 7.3: Multiplying Radical Expressions
Ex 10 Simplify.
β8 β543
β27 ββ300
β75π¦5 ββ32π63 β800
Ex 11 Multiply.
β6 β β5 β23
β β33
βπ₯ β π β βπ₯ + π β14 β β98
Ex 12 Find a simplified form of π(π₯) = β2π₯2 + 8π₯ + 8 and π(π₯) = β4π₯2 + 8π₯ + 8 Ex 13 Simplify. Assume that no radicands were formed by raising negative numbers to even powers.
a) βπ₯8π¦7 b) ββ32π7π11 5
c) β810π₯94 d) β2
3β43
Rules
βππ
β βππ
= βππ π
and βππ π
= βππ
β βππ
as long as βππ
and βππ
are real numbers.
FLC Ch 7
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e) (β10π₯2π¦43)(β20π₯2π¦63
) f) βπ3(π β π)45 βπ7(π β π)45
Ex 14 Simplify using the laws of exponents.
21/10 β 42/5 32/3 β 3β4/5
92/3 (9π2πβ4)
12
(1
2π₯π¦β1/3π§β1/2)
β6
(β8π₯2π¦7
)5
Express answer in exponential form Practice Problems-box ALL answers 1) Simplify. Assume that each variable can represent any real number.
a) β64π‘2 b) βπ2 + 14π + 49 c) β(π + 7)33
2) Write an equivalent expression using radical/exponential notation.
a) (β5ππ3
)4
b) (16π6)3/4
3) Simplify. Do not use negative exponents in answer.
a) (π₯β2/3)3/5
b) 7β1/3
7β1/2
4) Simplify. Write all answers in radical notation. Assume that all variables represent nonnegative numbers.
a) β250π₯3π¦2 b) β3π₯4π3
β ββ9π₯π23
FLC Ch 7
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Sec 7.4: Dividing Radical Expressions Ex 15 Simplify. Assume all variables represent positive numbers. AAVR+N a) b)
β25π5
π6 β
243π9
π15π4
5
Ex 16 Divide and if possible, simplify. AAVR+N
a) b)
β75ππ3
3β3
β64π11π285
β2ππβ25
Ex 17 Simplify. Assume all variables are nonnegative.
(β2)(β2) (β5)2
β92 (βπ₯)2
βπ₯33 β2π₯44
Ex 18 Rationalize each denominator. AAVR+N a) b)
3β5
2β7 β
2
9
3
Rule: For any real numbers βππ
and βππ
, π β 0, βπ
π
π=
βππ
βππ .
FLC Ch 7
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c) d)
β21π₯2π¦
β75π₯π¦5 β
7
64π2π4
4
Ex 19 Simplify. For all but the first two problems, write final answers in exponential form. AAVR+RN
β75ππ3
3β3 βπ₯5π¦π§76
1
9πβ8/9
8β7/9ππ₯β7/8π§8 62/3 β 63/4 (π₯2/3 π¦β4/5)1/2
Note: On quizzes and exams β must know when to use exponents vs radicals.
Sec 7.5: Expressions Containing Several Radical Terms
Ex 20 Add/subtract. Assume all variables represent nonnegative real numbers. AAVRNRN
a) β6 + 3β6 β 8β6 b) 5β12 + 16β27 c) β54π₯3
β β2π₯43
FLC Ch 7
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Ex 21 Multiply. AAVRNRN
a) 4β17(β2 β β17) b) βπ₯3
(β3π₯23β β81π₯23
)
c) (4β5 β 3β2)(2β5 + 4β2) d) Let π(π₯) = π₯2. Find π(4 β βπ₯ β 3).
Ex 22 Rationalize each denominator. a) PP b) c)
5
4 β β5
1 β β3
3 + β3
β2π₯
β2π₯ β βπ§
Ans: ππ+πβπ
ππ
d) e) f) Test?
π₯ β 36
βπ₯ β 6
3
β2π₯ + 5
βπ3 + π 33
βπ + π 3
Ex 23 Simplify. AAVRNRN
a) b) c) Provide 3 forms of the answer.
βπ43βπ34
βπ₯π¦2π§3
βπ₯3π¦π§2 βπ₯23
βπ₯65
exponential, radical not rationalized, rad ratl
FLC Ch 7
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Sec 7.6: Solving Radical Equations
**Always start by isolating the radical and check for extraneous solutions.** WILL NOT BE REMINDED TO ON EXAMS
Ex 24 Solve.
a) β2π₯ β 1 = 2 b) βπ₯ β 2 + 4 = 2 c) βπ₯ β 23
+ 4 = 7
d) 3π₯1/2 + 12 = 9 e) βπ¦3 = β4 f) β2π₯ + 34
β 5 = β2
g) π₯ = βπ₯ β 1 + 3 h) β2π‘ β 7 = β3π‘ β 12 i) β6π₯ + 7 β β3π₯ + 3 = 1
Defn A radical equation is an equation in which the variable appears in a radicand.
Examples: β2π₯3
+ 1 = 5 βπ β 2 = 7 4 β β3π₯ + 1 = β6 β π₯
The Principle of Powers If π = π, then ππ = ππ for any exponent π. Warning: The converse is not true.
FLC Ch 7
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Ex 25 If π(π₯) = βπ₯ + 6 + β2 β π₯, find any π₯ for which π(π₯) = 4. Show a check.
Sec 7.8: The Complex Numbers Ex Solve π₯2 β 1 = 0 and π₯2 + 1 = 0.
Ex 26 Express in terms of π.
ββ9 ββ7 β ββ75 4 β ββ60 ββ4 + ββ12
4ββ100 β1
2ββ20 + ββ27
3
Ex 27 Circle all irrational numbers. Box all nonreal, complex numbers. Double underline all rational numbers.
β12 β9 4π β7 ββ12 ββ83
β(β2)44
ββ4 + β9 2π ββ16 β(β2)33
β645
π
2+ ββ5
Defn of the Number π
π is the unique number for which π = ββ1 and π2 = β1. Note: π β βπ!!!
Defns An imaginary number is a number that can be written in the form π + ππ, where π and π are real numbers and π β 0. A complex number is any number that can be written in the form π + ππ where π and π are real numbers. Note: π and π can both be 0. The real part of a complex number is π. The imaginary part is π.
Conjugate of a Complex Number The conjugate of a complex number π + ππ is π β ππ, and the conjugate of π β ππ is π + ππ.
FLC Ch 7
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βCycle of πβ Ex 28 Perform the indicated operation and simplify. Write each answer in π + ππ form. Identify the real and imaginary parts.
a) (8 + 7π) β (2 + 4π) b) 7π(β8π) c) ββ6ββ7 d) (β4 + 5π)(3 β 4π) e) (1 + 2π)(1 β 2π) f) (3 + 2π)2 g) h) i) 4
7π
26
5 + π
6π + 3
3π
FLC Ch 7
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j) k) l) m) Assume π₯ β₯ 0
8 + 9π
9 β 3π π78 5π85 + 4π403 ββ18π₯ββ45π₯
Sec 7.7: The Distance and Midpoint Formulas and Other Applications Ex 29 (# 20) How long is a guy wire if it reaches from the top of a 15-ft pole to a point on the ground 10 ft from the pole?
The Principle of Square Roots
For any nonnegative real number π, if ππ = π then π = βπ or π = ββπ.
The Pythagorean Theorem
In any right triangle, if π and π are the lengths of the legs and π is the length of the hypotenuse,
then ππ + ππ = ππ.
π
π³ππ
π³ππ π―πππππππππ
π ππΒ°
π
Lengths Within Isosceles and 30Β° β 60Β° β 90Β° Right Triangles
The length of the hypotenuse in an The length of the longer leg in a 30Β° β 60Β° β 90Β° isosceles right triangle is the length right triangle is the length of the shorter leg times
βπ.
of a leg times βπ. The hypotenuse is twice as long as the shorter leg.
π
ππ πβπ
ππΒ°
ππΒ° π
πβπ π
ππΒ°
ππΒ°
FLC Ch 7
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Ex 30 For each triangle, find the missing length(s). Give an exact answer and, where appropriate, an approximation to 3 decimal places. a) (# 42) 18.385 b) (# 44) 13.435 Ex 31 (# 54) Find the distance between the pair of points (β1, β4) and (β3, β5). Do two ways. PT then DF.
Ex 32 (# 74) Find the midpoint of the segment with endpoints
(β4
5, β
2
3) πππ (
1
8,3
4).
The Distance Formula
The distance π between any two points (π₯1, π¦1) and (π₯2, π¦2) is given by
π = β(ππ β ππ)π + (ππ β ππ)π.
The Midpoint Formula
If the endpoints of a segment are (π₯1, π¦1) and (π₯2, π¦2), then the coordinates of the midpoint are
(ππ+ππ
π,
ππ+ππ
π). Note: To locate the midpoint, average the π₯-coordinates and average the π¦-coordinates.
(ππ, ππ)
(ππ, ππ)
(ππ + ππ
π,ππ + ππ
π)
?
?
?
?
ππ ππ ππ
ππ
ππ
?
FLC Ch 7
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Practice Problems (sec 7.2)
(125π₯1/2)2/3 (64π₯π¦12/5)
1/6
(8π₯2π¦3)1/3
(17π₯β2/5π¦1/4π§1/3)5
(51π₯4π¦β3/4π§β1/6)3 16π326 + 12π20
(Sec 6.5) Terrel bicycles 10 mph with no wind. Against the wind, he bikes 12 miles in the same amount of time that it takes him to bike 48 miles with the wind. Set up an equation or a system of equations to find the speed of the wind. Circle your equation(s). Next, solve and circle your answer.