Beginning and Intermediate Algebra - Wallace Math Courses ...

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Example 216. a3 Using the quotient rule, subtract exponents a5 a −2 Our Solution, but we will also solve this problem another way. a3 Rewrite exponents as repeated multiplication a5 aaa aaaaa Reduce three a′ s out of top and bottom 1 aa Simplify to exponents 1 Our Solution, putting these solutions together gives: a2 a−2 = 1 Our Final Solution a2 This example illustrates an important property of exponents. Negative exponents yield the reciprocal of the base. Once we take the reciprical the exponent is now positive. Also, it is important to note a negative exponent does not mean the expression is negative, only that we need the reciprocal of the base. Following are the rules of negative exponents Rules of Negative Exponets: a −m = 1 m 1 = am a−m � a �−m = b bm am Negative exponents can be combined in several different ways. As a general rule if we think of our expression as a fraction, negative exponents in the numerator must be moved to the denominator, likewise, negative exponents in the denominator need to be moved to the numerator. When the base with exponent moves, the exponent is now positive. This is illustrated in the following example. Example 217. a3b−2c 2d−1 e−4 Negative exponents on b, d, and e need to flip f2 a 3 cde 4 2b 2 f 2 Our Solution 184
As we simplified our fraction we took special care to move the bases that had a negative exponent, but the expression itself did not become negative because of those exponents. Also, it is important to remember that exponents only effect what they are attached to. The 2 in the denominator of the above example does not have an exponent on it, so it does not move with the d. We now have the following nine properties of exponents. It is important that we are very familiar with all of them. Properties of Exponents a m a n = a m+n (ab) m = a m b m a −m = 1 a m am = am−n an � a �m = b am bm (a m ) n = a mn a 0 = 1 1 = am a−m � a �−m = b bm am World View Note: Nicolas Chuquet, the French mathematician of the 15th century wrote 12 1m¯ to indicate 12x −1 . This was the first known use of the negative exponent. Simplifying with negative exponents is much the same as simplifying with positive exponents. It is the advice of the author to keep the negative exponents until the end of the problem and then move them around to their correct location (numerator or denominator). As we do this it is important to be very careful of rules for adding, subtracting, and multiplying with negatives. This is illustrated in the following examples Example 218. 4x −5 y −3 · 3x 3 y −2 6x −5 y 3 12x −2 y −5 6x −5 y 3 Simplify numerator with product rule, adding exponents Quotient rule to subtract exponets, be careful with negatives! ( − 2) − ( − 5) =(−2)+5=3 ( − 5) − 3=(−5)+(−3) = − 8 2x 3 y −8 Negative exponent needs to move down to denominator 2x 3 y 8 Our Solution 185
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Beginning and Intermediate Algebra
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Special thanks to: My beautiful wif
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Chapter 6: Factoring 6.1 Greatest C
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0.1 Pre-Algebra - Integers Objectiv
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Multiplication and division of inte
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45) (4)( − 6) Find each quotient.
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9 ÷ 3 3 = 21 ÷ 3 7 Our Soultion T
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Example 21. 13 6 − 9 6 4 6 2 3 Sa
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Find each quotient. 37) − 2 ÷ 7
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the expression that is to be evalua
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Solve. 1) − 6 · 4( − 1) 3) 3+(
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It will be more common in our study
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0.4 Practice - Properties of Algebr
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Chapter 1 : Solving Linear Equation
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Addition Problems To solve equation
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8x = − 24 8 8 x = − 3 Division
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1.2 Linear Equations - Two-Step Equ
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− 1 − 1 Divide both sides by
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1.3 Solving Linear Equations - Gene
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Example 63. − 3x +9=6x − 27 Not
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4[ − 2]+9=−15 +8(2) Finish mult
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1.4 Solving Linear Equations - Frac
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Example 72. � � 3 5 4 x + = 3 2
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1.5 Solving Linear Equations - Form
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Example 79. A = πr 2 +πrs for s S
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1.5 Practice - Formulas Solve each
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|x| = 5 Absolute value can be posit
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Example 90. 7+|2x − 5| =4 Notice
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1.7 Solving Linear Equations - Vari
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Once we have found the constant of
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1.7 Practice - Variation Write the
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sure. If the pressure of a certain
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Fifteen more than three times a num
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When we started with our first, sec
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1.8 Practice - Number and Geometry
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30. The perimeter of a college bask
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Our equation comes from the future
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4x + 2=102 Solve the two − step e
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years ago, the bronze plaque was on
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1.10 Solving Linear Equations - Dis
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Rate Time Distance Bob r + 2 3 Fred
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2 =t Our solution fort, she catches
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Find the distance he rode. 8. A man
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The first plan is flying 25 mph slo
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2.1 Graphing - Points and Lines Obj
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E B D F G C A Our Solution The main
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x y − 3 − 4 0 − 2 3 0 Our com
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2.2 Graphing - Slope Objective: Fin
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When mathematicians began working w
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Example 131. Find the value of x be
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9) 10) Find the slope of the line t
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y = − 2 x + 3 Our Solution 3 Iden
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2.3 Practice - Slope-Intercept Writ
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2.4 Graphing - Point-Slope Form Obj
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Find the equation of the line throu
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Write the point-slope form of the e
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m = 2 5 m = 2 5 Example 147. Slope
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2.5 Practice - Parallel and Perpend
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Chapter 3 : Inequalities 3.1 Solve
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than 4. If we have an expression su
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− 5 − 5 − 2x � 6 Divide bot
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Solve each inequality, graph each s
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As the graphs overlap, we take the
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3.2 Practice - Compound Inequalitie
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edition of a college algebra text!
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It is important to remember as we a
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Page 135 and 136: ested in the point that is a soluti
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Page 139 and 140: 4.2 Systems of Equations - Substitu
Page 141 and 142: tion. This process is described and
Page 143 and 144: World View Note: French mathematici
Page 145 and 146: 31) 2x + y = 2 3x + 7y = 14 33) x +
Page 147 and 148: 6x +9y = 6 New second equation −
Page 149 and 150: Just as with graphing and substutio
Page 151 and 152: 4.4 Systems of Equations - Three Va
Page 153 and 154: just the ordered pair (x, y). In th
Page 155 and 156: 15x − 12y +9z = − 12 5x − 4y
Page 157 and 158: 23) 3x + 3y − 2z = 13 6x + 2y −
Page 159 and 160: are given to use. This means someti
Page 161 and 162: − 0.5 − 0.5 c=17 We have c, num
Page 163 and 164: − 0.06x − 0.06y = − 240 Add e
Page 165 and 166: 14) A bank contains 27 coins in dim
Page 167 and 168: 4.6 Systems of Equations - Mixture
Page 169 and 170: Amount Part Total Start 40 3 120 Ad
Page 171 and 172: 4(30 −n)+2.5n=105 Substitute into
Page 173 and 174: 16) A certain grade of milk contain
Page 175 and 176: 175
Page 177 and 178: 5.1 Polynomials - Exponent Properti
Page 179 and 180: A quicker method to arrive at the s
Page 181 and 182: Example 210. Example 211. 7a 3 (2a
Page 183: 5.2 Polynomials - Negative Exponent
Page 187 and 188: 5.2 Practice - Negative Exponents S
Page 189 and 190: Example 223. Convert 3.21 × 10 5 t
Page 191 and 192: 5.3 Practice - Scientific Notation
Page 193 and 194: World View Note: Ada Lovelace in 18
Page 195 and 196: 21) (3+b 4 )+(7+2b + b 4 ) 22) (1+6
Page 197 and 198: Just as we distribute a monomial th
Page 199 and 200: Example 243. (2x − 5)(4x 2 − 7x
Page 201 and 202: 5.6 Polynomials - Multiply Special
Page 203 and 204: Be very careful when we are squarin
Page 205 and 206: 5.7 Polynomials - Divide Polynomial
Page 207 and 208: 3. Change the sign of the terms and
Page 209 and 210: Example 261. 2x 3 + 42 − 4x x + 3
Page 211 and 212: Chapter 6 : Factoring 6.1 Greatest
Page 213 and 214: ers using mental math, then we take
Page 215 and 216: 6.1 Practice - Greatest Common Fact
Page 217 and 218: Example 273. 10ab + 15b + 4a+6 Spli
Page 219 and 220: example the binomials are (a + b) a
Page 221 and 222: 6.3 Factoring - Trinomials where a
Page 223 and 224: As the past few examples illustrate
Page 225 and 226: 6.3 Practice - Trinomials where a =
Page 227 and 228: Example 296. 10x 2 − 27x +5 Multi
Page 229 and 230: 6.5 Factoring - Factoring Special P
Page 231 and 232: Example 306. 4x 2 + 20xy + 25y 2 Mu
Page 233 and 234: 6.5 Practice - Factoring Special Pr
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Example 313. Example 314. Example 3
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6.7 Factoring - Solve by Factoring
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0=(2x +3)(2x − 3) One factor must
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6.7 Practice - Solve by Factoring S
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7.1 Rational Expressions - Reduce R
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However, if there is more than just
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41) 8m+16 20m − 12 43) 2x2 − 10
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Example 333. 25x2 24y4 · 9y8 55x7
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Simplify each expression. 1) 8x2 9
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7.3 Rational Expressions - Least Co
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As the above example illustrates, w
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7.4 Rational Expressions - Add & Su
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13 12 Our Solution The same process
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7.4 Practice - Add and Subtract Add
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2(12) 1(12) − 3 4 5(12) 1(12) + 6
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LCD =(x + 3)(x − 3) Multiply each
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23) 25) 27) 29) x − 4+ 9 2x +3 x
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(20)(9)=6x Multiply 180 = 6x Divide
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3.5 6 The man hasashadow of 3.5 fee
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33) Kali reduced the size of a pain
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LCD will be more involved. We will
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x=5 or − 2 Our Solution World Vie
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7.8 Rational Expressions - Dimensio
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� 435 1 �� 1 lbs = 16 435
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To focus on the process of conversi
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7.8 Practice - Dimensional Analysis
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Chapter 8 : Radicals 8.1 Square Roo
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√ process is being able to transl
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Simplify. √ 1) 245 √ 3) 36 √
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Example 384. 3√ 54 We are working
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8.3 Radicals - Adding Radicals Obje
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Simiplify 1) 2 5 √ + 2 5 √ + 2
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√ √ √ √ 4 50 + 6 30 − 8 3
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The previous example could have bee
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8.5 Radicals - Rationalize Denomina
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2x2 √ √ 10 − 6x 3x Our Soluti
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Example 408. 2 5 √ − 3 7 √ 5
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37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
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1 (3) 4 Evaluate exponent 1 81 Our
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8.6 Practice - Rational Exponents W
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Example 420. ab2 3√ a2 4√ b Rew
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8.7 Practice - Radicals of Mixed In
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Example 426. Example 427. i 35 Divi
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√ Dividing with complex numbers a
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Simplify. 1) 3 − ( − 8+4i) 3) (
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Chapter 9 : Quadratics 9.1 Solving
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Example 444. + 1 +1 Add1to both sid
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2 − 2=0 Subtract 0=0 True! It wor
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Solve. √ 1) 2x + 3 √ 3) 6x −
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x4 4√ = ± 4√ 16 Simplify roots
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5√ 4x +1 = ± 3 Clear radical by
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9.3 Quadratics - Complete the Squar
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x2 + 5 25 x+ 3 36 � x + 5 �2 6
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Example 464. − 7 3 � 3 3 + 3 3
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9.4 Quadratics - Quadratic Formula
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x = Example 468. √ 30 ± 900 + 11
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9.4 Practice - Quadratic Formula So
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12x 2 − 17x + 6=0 Our Solution If
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9.5 Practice - Build Quadratics fro
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Example 479. a −2 − a −1 −
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12 = (x − 3) 2 or 13 =(x − 3) 2
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9.7 Quadratics - Rectangles Objecti
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Example 486. 2 Our Solution The len
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4(x 2 + 10x − 24) =0 Factor trino
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that require 24 square meters for p
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1(12x) 3 + 1(12x) x 1 1 5 + = 3 x 1
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6x +6(x − 1)=5x(x − 1) Reduce e
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16) A sink can be filled from the f
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These simultaneous product equation
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9.10 Quadratics - Revenue and Dista
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96n(n − 2) 96n(n − 2) +4n(n −
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upstream, the current will pull aga
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12) A pilot flying at a constant ra
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The above method to graph a parabol
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It is important to remember the gra
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Chapter 10 : Functions 10.1 Functio
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Example 504. Which of the following
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2x − 3 � 0 Set up an inequality
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Solve. 10.1 Practice - Function Not
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10.2 Functions - Operations on Func
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(x − 5)(x +1) (x − 5) Divide ou
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We can also evaluate a composition
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23) f(x) = x 2 − 5x g(x) = x + 5
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10.3 Functions - Inverse Functions
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(3x +6)(3 − 4x) 3 − 4x (4x+8)(3
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10.3 Practice - Inverse Functions S
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Example 530. 8 3x = 32 Rewrite 8 as
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10.4 Practice - Exponential Functio
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log3x =7 Identify base, 3, answer,
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10.5 Practice - Logarithmic Functio
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A = 25000(1.01625) 20 Evaluate expo
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The variable e is a constant simila
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j. All of the above compounded cont
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Using the diagram at right, find ea
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10.7 Practice - Trigonometric Funct
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21) 23) 25) 27) A B A 5 38 ◦ x x
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37) 39) A A 37.1 ◦ C x x C 13.1 4
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θ 17 From angle θ the given sides
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function to find the missing angle
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Find the measure of each angle indi
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27) 29) 14 A A θ C 15 7 C 14 θ B
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437
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7) 5 4 8) 4 3 9) 3 2 10) 8 3 11) 5
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54) 60v − 7 55) − 3x + 8x 2 56)
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45) 12 46) All real numbers 47) No
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15) wx 3 = 1458 16) h = 1.5 j 17) a
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1) B(4, − 3) C(1, 2) D( − 1, 4)
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1) y = 2x +5 2) y = − 6x +4 3) y
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23) x =4 24) y − 4 = 7 (x − 1)
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25) 5�x
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1) ( − 2, 4) 2) (2,4) 3) No solut
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27) 56, 144 28) 1.5, 3.5 29) 30 30)
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42) 1.2 × 10 6 5.4 1) 3 2) 7 3)
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5) 2x 3 + 4x 2 + x 2 6) 5p3 4 +4p2
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7) (b +8)(b + 4) 8) (b − 10)(b
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11) n(n − 1) 12) (5x +3)(x − 5)
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5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r
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7) 8) 5 24r 7x + 3y x 2 y 2 9) 15t+
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21) 0, 5 22) − 2, 5 3 23) 4,7 24)
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42) − 18xz 4x3yz3 � 4 8.3 1) 6
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16 3 √ + 4 5 √ 43 18) 19) 2 √
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8.8 1) 11 − 4i 2) − 4i 3) − 3
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√ √ 33) 4+i 39, 4 −i 39 34)
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9) ± 2, ± 4 10) 2,3, − 1 ±i 3
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1) 2) 3) 4) 5) 6) (-2,0) (0,-8) (-1
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11) 100 12) − 74 13) 1 5 14) 27 1
Page 487 and 488:
10.5 1) 9 2 = 81 2) b −16 = a 3)
Page 489:
21) 51.3 ◦ 22) 45 ◦ 23) 56.4
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