𝑖 𝑥 √ ,√ / , 0 - los rios community college...
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FLC Ch 7
Page 1 of 14
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.
√1
√49
√−81
√625
√83
√−273
√−13
√164
√−164
√130
−√180
5
−√225
√−225
√−1
32
5
Ex 3 Simplify. (Assume all variables represent any real number.) AAVRARN
√16𝑡2 √(𝑎 + 3)2 √𝑦33 √𝑦88
√(7𝑏)44 √7𝑏99
√(−4)66 √(−4)77
√9 − 6𝑦 + 𝑦2
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
√25𝑡2 −√(7𝑥𝑦)2 −√(7𝑥𝑦)4
−√64𝑦63 √𝑎14 √(𝑥 + 3)10
√𝑟𝑎𝑑𝑖𝑐𝑎𝑛𝑑𝑖𝑛𝑑𝑒𝑥
√ = √2
, √3
, √4
, etc.
√𝑚𝑢𝑠𝑡 𝑏𝑒 ≥ 0𝑒𝑣𝑒𝑛
√𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑑𝑑
√( )𝑛𝑛= {
| |, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛( ), 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
FLC Ch 7
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Ex 5 Find the domain of each function.
Evaluate: 𝑓(0), 𝑓(1), and 𝑓(−2)
Sec 7.2: Rational Numbers and Exponents
What does 𝒙𝟏/𝟐 mean? 𝒙𝟏/𝟑?
Examples:
91/2 = (32)1/2 = 31 = 3 81/3
161/2 271/3
491/2 1251/3
Recall: Exponential Rules
𝑥𝑎 ∙ 𝑥𝑏 = 𝑥𝑎+𝑏 𝑥𝑎
𝑥𝑏 = 𝑥𝑎−𝑏 , 𝑥 ≠ 0 (𝑥𝑎)𝑏 = 𝑥𝑎𝑏
(𝑥𝑦)𝑎 = 𝑥𝑎𝑦𝑎 (𝑥
𝑦)
𝑎=
𝑥𝑎
𝑦𝑎 𝑥−𝑎 =1
𝑥𝑎 , 𝑥 ≠ 0
𝑥0 = 1, 𝑥 ≠ 0
Conclusion: ( )1/2 = √ and ( )1/3 = √3
In general, 𝒂𝟏/𝒏 = √𝒂𝒏 . Note: If 𝒏 is even, 𝒂 must be ≥ 𝟎. If 𝒏 is odd, 𝒂 can be anything.
𝑎𝑚/𝑛 = √𝑎𝑚𝑛= ( √𝑎
𝑛)
𝑚 if √𝑎
𝑛 exists.
𝑓(𝑥) = √4 + 3𝑥
𝑔(𝑥) = √4 + 3𝑥3
ℎ(𝑥) = √4 + 3𝑥6
𝑠(𝑥) = 13𝑥 − √4 + 3𝑥
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?
𝑡1/4 (𝑎2𝑏)1/5 47/2 (9𝑦6)3/2
√104
√22 √𝑛45 √𝑐𝑑
3
Ex 7 Rewrite but do not simplify. (√2𝑥𝑦2𝑧3
)5 √𝑥2𝑦5𝑧73
Ex 8 Simplify. Do not use negative exponents in answer. Which numbers are rational? Irrational?
51/4 ∙ 51/8 87/11
8−2/11 (55/4)
3/7 (274/9𝑥−1/3𝑦2/5)
3/2
Ex 9 Simplify. Present answers in radical form.
√𝑎312 (√𝑎𝑏
3)
15 √(3𝑥)28
√𝑎𝑏153
√√𝑥6
(√𝑥2𝑦53)
12 √√2𝑎
35
Several problems from “Writing Expressions as Powers of ‘x’” handout page 38.
FLC Ch 7
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Sec 7.3: Multiplying Radical Expressions
Ex 10 Simplify.
√8 √543
√27 −√300
√75𝑦5 √−32𝑎63 √800
Ex 11 Multiply.
√6 ∙ √5 √23
∙ √33
√𝑥 − 𝑎 ∙ √𝑥 + 𝑎 √14 ∙ √98
Ex 12 Find a simplified form of 𝑓(𝑥) = √2𝑥2 + 8𝑥 + 8 and 𝑔(𝑥) = √4𝑥2 + 8𝑥 + 8 Ex 13 Simplify. Assume that no radicands were formed by raising negative numbers to even powers.
a) √𝑥8𝑦7 b) √−32𝑎7𝑏11 5
c) √810𝑥94 d) √2
3√43
Rules
√𝑎𝑛
∙ √𝑏𝑛
= √𝑎𝑏 𝑛
and √𝑎𝑏 𝑛
= √𝑎𝑛
∙ √𝑏𝑛
as long as √𝑎𝑛
and √𝑏𝑛
are real numbers.
FLC Ch 7
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e) (√10𝑥2𝑦43)(√20𝑥2𝑦63
) f) √𝑎3(𝑏 − 𝑐)45 √𝑎7(𝑏 − 𝑐)45
Ex 14 Simplify using the laws of exponents.
21/10 ∙ 42/5 32/3 ∙ 3−4/5
92/3 (9𝑘2𝑚−4)
12
(1
2𝑥𝑦−1/3𝑧−1/2)
−6
(√8𝑥2𝑦7
)5
Express answer in exponential form Practice Problems-box ALL answers 1) Simplify. Assume that each variable can represent any real number.
a) √64𝑡2 b) √𝑐2 + 14𝑐 + 49 c) √(𝑐 + 7)33
2) Write an equivalent expression using radical/exponential notation.
a) (√5𝑎𝑏3
)4
b) (16𝑎6)3/4
3) Simplify. Do not use negative exponents in answer.
a) (𝑥−2/3)3/5
b) 7−1/3
7−1/2
4) Simplify. Write all answers in radical notation. Assume that all variables represent nonnegative numbers.
a) √250𝑥3𝑦2 b) √3𝑥4𝑏3
∙ √−9𝑥𝑏23
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)
√25𝑎5
𝑏6 √
243𝑎9
𝑏15𝑐4
5
Ex 16 Divide and if possible, simplify. AAVR+N
a) b)
√75𝑎𝑏3
3√3
√64𝑎11𝑏285
√2𝑎𝑏−25
Ex 17 Simplify. Assume all variables are nonnegative.
(√2)(√2) (√5)2
√92 (√𝑥)2
√𝑥33 √2𝑥44
Ex 18 Rationalize each denominator. AAVR+N a) b)
3√5
2√7 √
2
9
3
Rule: For any real numbers √𝑎𝑛
and √𝑏𝑛
, 𝑏 ≠ 0, √𝑎
𝑏
𝑛=
√𝑎𝑛
√𝑏𝑛 .
FLC Ch 7
Page 7 of 14
c) d)
√21𝑥2𝑦
√75𝑥𝑦5 √
7
64𝑎2𝑏4
4
Ex 19 Simplify. For all but the first two problems, write final answers in exponential form. AAVR+RN
√75𝑎𝑏3
3√3 √𝑥5𝑦𝑧76
1
9𝑝−8/9
8−7/9𝑎𝑥−7/8𝑧8 62/3 ∙ 63/4 (𝑥2/3 𝑦−4/5)1/2
Note: On quizzes and exams – 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) √6 + 3√6 − 8√6 b) 5√12 + 16√27 c) √54𝑥3
− √2𝑥43
FLC Ch 7
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Ex 21 Multiply. AAVRNRN
a) 4√17(√2 − √17) b) √𝑥3
(√3𝑥23− √81𝑥23
)
c) (4√5 − 3√2)(2√5 + 4√2) d) Let 𝑓(𝑥) = 𝑥2. Find 𝑓(4 − √𝑥 − 3).
Ex 22 Rationalize each denominator. a) PP b) c)
5
4 − √5
1 − √3
3 + √3
√2𝑥
√2𝑥 − √𝑧
Ans: 𝟐𝟎+𝟓√𝟓
𝟏𝟏
d) e) f) Test?
𝑥 − 36
√𝑥 − 6
3
√2𝑥 + 5
√𝑟3 + 𝑠33
√𝑟 + 𝑠3
Ex 23 Simplify. AAVRNRN
a) b) c) Provide 3 forms of the answer.
√𝑏43√𝑏34
√𝑥𝑦2𝑧3
√𝑥3𝑦𝑧2 √𝑥23
√𝑥65
exponential, radical not rationalized, rad ratl
FLC Ch 7
Page 9 of 14
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) √2𝑥 − 1 = 2 b) √𝑥 − 2 + 4 = 2 c) √𝑥 − 23
+ 4 = 7
d) 3𝑥1/2 + 12 = 9 e) √𝑦3 = −4 f) √2𝑥 + 34
− 5 = −2
g) 𝑥 = √𝑥 − 1 + 3 h) √2𝑡 − 7 = √3𝑡 − 12 i) √6𝑥 + 7 − √3𝑥 + 3 = 1
Defn A radical equation is an equation in which the variable appears in a radicand.
Examples: √2𝑥3
+ 1 = 5 √𝑎 − 2 = 7 4 − √3𝑥 + 1 = √6 − 𝑥
The Principle of Powers If 𝑎 = 𝑏, then 𝑎𝑛 = 𝑏𝑛 for any exponent 𝑛. Warning: The converse is not true.
FLC Ch 7
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Ex 25 If 𝑔(𝑥) = √𝑥 + 6 + √2 − 𝑥, find any 𝑥 for which 𝑔(𝑥) = 4. Show a check.
Sec 7.8: The Complex Numbers Ex Solve 𝑥2 − 1 = 0 and 𝑥2 + 1 = 0.
Ex 26 Express in terms of 𝑖.
√−9 √−7 − √−75 4 − √−60 √−4 + √−12
4√−100 −1
2√−20 + √−27
3
Ex 27 Circle all irrational numbers. Box all nonreal, complex numbers. Double underline all rational numbers.
√12 √9 4𝑒 √7 √−12 √−83
√(−2)44
√−4 + √9 2𝜋 √−16 √(√2)33
√645
𝜋
2+ √−5
Defn of the Number 𝑖
𝑖 is the unique number for which 𝑖 = √−1 and 𝑖2 = −1. Note: 𝒊 ≠ −𝟏!!!
Defns An imaginary number is a number that can be written in the form 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are real numbers and 𝑏 ≠ 0. A complex number is any number that can be written in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers. Note: 𝑎 and 𝑏 can both be 0. The real part of a complex number is 𝑎. The imaginary part is 𝑏.
Conjugate of a Complex Number The conjugate of a complex number 𝑎 + 𝑏𝑖 is 𝑎 − 𝑏𝑖, and the conjugate of 𝑎 − 𝑏𝑖 is 𝑎 + 𝑏𝑖.
FLC Ch 7
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“Cycle of 𝑖” Ex 28 Perform the indicated operation and simplify. Write each answer in 𝑎 + 𝑏𝑖 form. Identify the real and imaginary parts.
a) (8 + 7𝑖) − (2 + 4𝑖) b) 7𝑖(−8𝑖) c) √−6√−7 d) (−4 + 5𝑖)(3 − 4𝑖) e) (1 + 2𝑖)(1 − 2𝑖) f) (3 + 2𝑖)2 g) h) i) 4
7𝑖
26
5 + 𝑖
6𝑖 + 3
3𝑖
FLC Ch 7
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j) k) l) m) Assume 𝑥 ≥ 0
8 + 9𝑖
9 − 3𝑖 𝑖78 5𝑖85 + 4𝑖403 √−18𝑥√−45𝑥
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 𝑛, if 𝒙𝟐 = 𝒏 then 𝒙 = √𝒏 or 𝒙 = −√𝒏.
The Pythagorean Theorem
In any right triangle, if 𝑎 and 𝑏 are the lengths of the legs and 𝑐 is the length of the hypotenuse,
then 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐.
𝒄
𝑳𝒆𝒈
𝑳𝒆𝒈 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒂 𝟗𝟎°
𝒃
Lengths Within Isosceles and 30° − 60° − 90° Right Triangles
The length of the hypotenuse in an The length of the longer leg in a 30° − 60° − 90° isosceles right triangle is the length right triangle is the length of the shorter leg times
√𝟑.
of a leg times √𝟐. The hypotenuse is twice as long as the shorter leg.
𝒂
𝟐𝒂 𝒂√𝟑
𝟑𝟎°
𝟔𝟎° 𝒂
𝒂√𝟐 𝒂
𝟒𝟓°
𝟒𝟓°
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 (−1, −4) and (−3, −5). Do two ways. PT then DF.
Ex 32 (# 74) Find the midpoint of the segment with endpoints
(−4
5, −
2
3) 𝑎𝑛𝑑 (
1
8,3
4).
The Distance Formula
The distance 𝑑 between any two points (𝑥1, 𝑦1) and (𝑥2, 𝑦2) is given by
𝒅 = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐.
The Midpoint Formula
If the endpoints of a segment are (𝑥1, 𝑦1) and (𝑥2, 𝑦2), then the coordinates of the midpoint are
(𝒙𝟏+𝒙𝟐
𝟐,
𝒚𝟏+𝒚𝟐
𝟐). Note: To locate the midpoint, average the 𝑥-coordinates and average the 𝑦-coordinates.
(𝒙𝟏, 𝒚𝟏)
(𝒙𝟐, 𝒚𝟐)
(𝒙𝟏 + 𝒙𝟐
𝟐,𝒚𝟏 + 𝒚𝟐
𝟐)
?
?
?
?
𝟏𝟗 𝟏𝟑 𝟏𝟑
𝟏𝟑
𝟏𝟑
?
FLC Ch 7
Page 14 of 14
Practice Problems (sec 7.2)
(125𝑥1/2)2/3 (64𝑥𝑦12/5)
1/6
(8𝑥2𝑦3)1/3
(17𝑥−2/5𝑦1/4𝑧1/3)5
(51𝑥4𝑦−3/4𝑧−1/6)3 16𝑖326 + 12𝑖20
(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.