Algebra demystified (MGH, 2003)(ISBN 0071389938)(454s)_MAt

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Algebra demystified (MGH, 2003)(ISBN 0071389938)(454s)_MAt_.pdf

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Date Apr 6, 2006

Solutions
1: 2: 3: 4: 5: 4 1 4À1 3 À¼ ¼ 77 7 7 1 3 1þ3 4 þ¼ ¼ 55 5 5 1 1 1þ1 2 1 þ¼ ¼¼ 66 6 63 5 1 5À1 4 1 À¼ ¼ ¼ 12 12 12 12 3 2 9 2 þ 9 11 þ¼ ¼ ¼1 11 11 11 11...

CHAPTER 1 Fractions
À ÁÀ Á À Á 4 2 þ 3 1 ¼ 4 þ 2 þ 3 þ 1 ¼ ð4 þ 3Þ þ 2 þ 1 3 2 3 2 3 2 À Á ¼ 7 þ 4 þ 3 ¼ 7 þ 7 ¼ 7 þ 1 þ 1 ¼ 81 6 6 6 6 6 The other way is to convert the mixed numbers into improper fractions then add. 42 þ 31 ¼ 3 2 14 7 28 21 49 þ¼ þ¼ ¼ 81 6 32 6 6 6...

Practice
1. Tony is three years older than Marie. Marie’s age ¼ ________ Tony’s age ¼ ________ 2. Sandie is three-fourths as tall as Mona. Mona’s height (in the given unit of measure) ¼ ________ Sandie’s height (in the given unit of measure) ¼ ________ 3. Michael takes two hours longer than Gina to compute his taxes. Number of hours Gina takes to compute her taxes ¼ ________ Number of hours Michael takes to compute his taxes ¼ ________ 4. Three-ﬁfths of a couple’s net income is spent on rent Net income ¼ ________ Amount spent on rent ¼ ________ 5. A rectangle’s length is four times its width Width (in the given unit of measure) ¼ ________ Length (in the given unit of measure) ¼ ________ 6. Candice paid $5000 last year in federal and state income taxes. Amount paid in federal income taxes ¼ ________ Amount paid in state income taxes ¼ ________...

Chapter Review
3 6x 1 x 1 1 ðaÞ ðbÞ ðcÞ ðdÞ 2 2 x 2x 7x 2. Rewrite as a product of a number and a variable: 15 7 71 7 7 x ðbÞ Á ðcÞ þx ðdÞ Àx ðaÞ 15 15 x 15 15 1. Reduce to lowest terms: 3. x1 À¼ 43 xÀ1 ðaÞ 12...

Examples
À3ð5xÞ ¼ À15x À12ðÀ4xÞ ¼ 48x À2xð3yÞ ¼ À6xy À16xðÀ4yÞ ¼ 64xy À3ðÀxÞ ¼ 3x 5ðÀxÞ ¼ À5x ÀxðÀyÞ ¼ xy xðÀyÞ ¼ Àxy 4ðÀ1:83xÞð2:36yÞ ¼ À17:2752xy...

The expression 4x is shorthand for x þ x þ x þ x, that is x added to itself four times. Likewise x4 is shorthand for x Á x Á x Á x—x multiplied by itself four times. In x4 , x is called the base and 4 is the power or exponent. We say ‘‘x to the fourth power’’ or simply ‘‘x to the fourth.’’ There are many useful exponent properties. For the rest of the chapter, a is a nonzero number. Property 1 anam ¼ am+n...

Examples
43 4 3x 4 3x 4 À 3x À ¼ 2À Á ¼ 2À 2 ¼ 2 xx xx x x x x2 13 6 13 z 6 xy 13z 6xy 13z À 6xy ¼ 2À 2¼ À ¼ 2Á À Á 2 yz xy z yz xy xy z xy z xy xy2 z...

Examples
pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ pﬃﬃﬃﬃﬃ 64 ¼ 4 Á 16 ¼ 4 Á 16 ¼ 2 Á 4 ¼ 8 pﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 5 5 4 6x 4 4y ¼ 4 24xy 3 4x ¼ 12x Property 1 only applies to multiplication. There is no similar property for addition (nor subtraction). A cﬃﬃﬃmﬃmﬃon mistake is to ‘‘simplify’’ the sum of pﬃo ﬃﬃﬃ ﬃﬃﬃ ﬃ two squares. For example x2 þ 9 ¼ x þ 3 is incorrect. The following example should give you an idea ﬃoﬃfﬃﬃﬃwﬃhy pﬃese pﬃﬃ o expressions are not thﬃ tw p ﬃ ﬃ ﬃﬃﬃ n n equal. If there were the property a þ b ¼ a þ n b, then we would have pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃ 58 ¼ 49 þ 9 ¼ 49 þ 9 ¼ 7 þ 3 ¼ 10: This could only be true if 102 = 58. rﬃﬃ pﬃﬃ n a na ¼ pﬃﬃ Property 2 n b b We can take the quotient then the root or the individual roots then the quotient. rﬃﬃ pﬃﬃ 42 4 ¼ pﬃﬃ ¼ 9 93 Property 3 negative.) ﬃﬃ Àpﬃﬃ Ám pﬃﬃﬃm n a ¼ na (Remember that if n is even, then a must not be...

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 5 5 5 5 5 64x6 y3 ¼ 25 Á 2x5 xy3 ¼ 25 x5 2xy3 ¼ 2x 2xy3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 3 3 3 ð2x À 7Þ5 ¼ ð2x À 7Þ3 ð2x À 7Þ2 ¼ ð2x À 7Þ3 ð2x À 7Þ2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 ¼ ð2x À 7Þ ð2x À 7Þ2...

Practice
3 1: pﬃﬃ ¼ 5 7 2: pﬃﬃ ¼ y rﬃﬃ 6 ¼ 3: 7 8x 4: pﬃﬃ ¼ 3 rﬃﬃﬃﬃﬃﬃﬃ 7 xy 5: ¼ 11...

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃ ﬃﬃ ﬃﬃ ﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃ3ﬃﬃﬃ2ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃﬃﬃ2ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃﬃﬃ2ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃﬃﬃ2ﬃ 8 8 8 8 12 8 x y 12x y 12x y 12x y 8 12 8: ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Á pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 8 8 xy 8 x5 y6 x5 y6 8 x3 y2 x8 y8 ðxyÞ8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 4 4 4 1 1 2x3 y2 2x3 y2 2x3 y2 2x3 y2 9: pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Á pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 4 4 4 2xy 4 8xy2 23 x1 y2 4 2x3 y2 24 x4 y4 ð2xyÞ4 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ 5 5 5 5 5 4 4 32 x2 y4 4 Á 9x2 y4 36x2 y4 36x2 y4 5 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Á pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 10: 5 5 3xy 5 27x3 y 33 x3 y1 5 32 x2 y4 35 x5 y5 ð3xyÞ5...

CHAPTER 5 Exponents and Roots
3x þ 8 3x þ 8 7: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ¼ ð3x þ 8Þð12x þ 5ÞÀ3=7 7 3 ð12x þ 5Þ3=7 ð12x þ 5Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xÀ3 x À 3 ðx À 3Þ1=2 ¼ pﬃﬃﬃﬃ ¼ 8: ¼ ðx À 3Þ1=2 yÀ5=2 5 y5 y5=2 y sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 3 4 16x ð16x3 Þ1=4 16x3 9: ¼ ð16x3 Þ1=4 ð3x þ 1ÞÀ1=4 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 4 3x þ 1 3x þ 1 ð3x þ 1Þ1=4 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 4 ðx À 1Þ4 ðx À 1Þ4=5 5 ðx À 1Þ 10: ¼ ðx À 1Þ4=5 ðx þ 1ÞÀ3=5 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 5 ðx þ 1Þ3 3 ðx þ 1Þ3=5 ðx þ 1Þ One of the uses of these expﬃﬃnﬃent-root properties is to simplify multiple po ﬃﬃ roots. Using the properties n am ¼ am=n and ðam Þn ¼ amn , gradually rewrite the multiple roots as an exponent then as a single root....

Examples
7ðx À yÞ ¼ 7x À 7y x2 ð3x À 5yÞ ¼ 3x3 À 5x2 y 6x2 y3 ð5x À 2y2 Þ ¼ 30x3 y3 À 12x2 y5 4ð3x þ 1Þ ¼ 12x þ 4 8xyðx3 þ 4yÞ ¼ 8x4 y þ 32xy2 pﬃﬃﬃ 2 pﬃﬃﬃ pﬃﬃﬃ xðx þ 12Þ ¼ x2 x þ 12 x...

Practice
1: 2: 3: 4: 5: 7 4 þ ¼ 2x þ 3 x À 2 1 x þ ¼ xÀ1 xþ2 3x À 4 À2¼ xþ5 x y þ ¼ 2x þ y 3x À 4y x x þ ¼ 6x þ 3 6x À 3...