University:
California State University, Northridge
Course:
MATH 102 | Pre-Calculus I
Academic year:

2002
Views:

242

Pages:

19
Author:

Mettandc

Laws of Exponents Addition of Exponents If: a ≠ 0, a m • a n = a m+n Example: 23 • 22 = (2 • 2 • 2) • (2 • 2) = 2•2•2•2•2 = 25 = 23+2 31 • 35 = ? Another Example For Negative Exponents 2-3 • 2-2 = = = 1 1 • 3 2 = 2 2 1 23 • 22 1 2 3+2 = 1 25 = 2-5 = 2-3+-2 3-1 • 3-4 = ? Still More Examples 23 • 2-2 = 23+(–2) = 21 = 2 Why? 23 • 2-2 = 23 • 1 22 = 23 22 = 2• 2 •2 2• 2 = 2• 2 •2 2•2 = 1 • 2 = 2 = 21 More Exponents am m-n = a an If a ≠ 0 Example: 27 3 = 2 So, 2•2• 2 •2•2•2 • 2 2 •2•2 = 2•2• 2 •2•2•2•2 2•2•2 = 1 • 24 = 24 (=16) 27 3 = 2 27-3 = 24 More Exponents Example: 32 5 = 3 = = = So, 32 5 = 3 3•3 3• 3•3•3•3 3•3 1 • 3• 3 3•3• 3 1 1• 3 3  1 1 -3 3 = 3 =   3 27  32-5 = 3-3 Example: So, 45 5 = 4 = 4 •4• 4•4• 4 4 •4• 4•4• 4 1 45 5 = 4 45-5 = 40 = 1 If a and b are not zero, (ab )n = a n b n Example: (3 • 5)4 = (3 • 5) (3 • 5) (3 • 5) (3 • 5) = 3•3•3•3•5•5•5•5 = 34 • 54 (= 81 x 625 = 50,625) Example: (2 • 5)3 = (2 • 5) (2 • 5) (2 • 5) = 2•2•2•5•5•5 = 23 • 53 (= 1,000) Note that (2 • 5)3 = 103 = 1,000 So we have just "factored" 1,000  a n an   = n  b b Example:  3 4 3 3 3 3   = • • •  5 5 5 5 5 34 = 4 5 If a ≠ 0 (a m)n = a mn Example: (2 3 )4 = 2 3 • 2 3 • 2 3 • 2 3 = 2 3+ 3+3 + 3 = 23•4 = 212 (= 4096) Example:  1 2 3 1 1  3    =  •   3 3  3   1 1  1 1  1 1  =  •  •  •   3 3  3 3  3 3   1 6 =    3 So,  1 2 3  1 2 x3    =    3  3   1 6 =    3  1  =   729  Practice Simplify each expression (you may leave answers in exponential form.) a) 23 • 28  2 4 b)   • 5  5 56 c) 4 5 d) 23 • 32 x 20 e) x 20 f) 7-2 • 7-4 g) (24) 7 h) 30 i) (x2y) 4 Challenge: j)  1  −3    2 Answers to Practice Problems a) 2 3 • 2 8 = 2 3+8 = 211  2 4 24 1 24 b)   • 5 = 4 • 5 = 3  5 5 5 56 c) 4 = 56 −4 = 52 = 25 5 d) 2 3 • 32 = 23 • 32 (bases are different) x 20 20 −20 0 e) = x = x =1 20 x f) 7 −2 • 7 −4 = 7 −2+−4 = 7 −6 = g) (2 ) 4 7 1 76 = 2 28 h) 30 = 1 i) (x j)  1  −3   = (2 −1 ) −3 = 2 −1•−3 = 2 3 = 8  2 2 • y ) 4 = x8 y 4 Definitions • Square Root - a number that when multiplied by itself produces a given number. Example: • Radical Sign 3 is a square root of 9. -3 is also a square root of 9. The symbol for the positive square root. By convention, the radical sign, a designates only the positive square root. Radical Sign → 36 ← Radicand Squaring and Taking a Square Root are Inverse Operations Like Addition is to Subtraction Like Multiplication is to Division 52 = 25 and 25 = 5 8 2 = 64 and 64 = 8 Square Numbers and Square Roots Square Numbers 12 1 2 2 4 32 9 42 16 2 5 25 2 6 36 72 49 82 64 2 9 81 2 10 100 Pos. Square Roots 1 2 3 4 5 6 7 8 9 10 Perspective When taking a Square Root, one question you could ask yourself is: "What number times itself is the radicand?” 36 = ? "What number times itself is 36?" Practice Taking the Square Root 1. 49 2. 121 3. 100 4. 1 5. 225 6. 81 7. 169 8. 36 9. 289 10. 144 Answers to Practice 1. 49 = 7 2. 121 = 11 3. 100 = 10 4. 1 = 1 5. 225 = 15 6. 81 = 9 7. 169 = 13 8. 36 = 6 9. 289 = 17 10. 144 = 12 Question Between which whole numbers is the Square Root of 45? 45 45 is not a square number, but it is between two square numbers. 36 < 45 < 49 36 < 45 < 49 and 6 < and 45 < 7 45 is a little less than 7. ( 45 ≈ 6.708) Practice Estimate the value of 67 Practice Estimate the value of 67 Since 64 < 67 < 81 and 64 < 67 < 81 and 8 < 67 < 9 67 is a little more than 8. 67 ≈ 8.185