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NAME_____________________DATE_____________PERIOD_______ Exponents Exponents are a way to show how many times something is multiplied by itself. The small superscript number is the number of times you multiply the number below it by itself Examples: 42 = 4 x 4 = 16 43 = 4 x 4 x 4 = 64 28 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 Laws of Exponents (Always write as positive exponent) Law x1 = x x0 = 1 x-n = 1/xn xmxn = xm+n xm/xn = xm-n (xm)n = xmn x-n = 1/xn Example 61 = 6 70 = 1 4-2 = 1/42 2223 = 22+3 = 25 46/42 = 46-2 = 44 (32)3 = 32×3 = 36 5-3 = 1/53 a) 35 ______ b) (-2)4 ______ c) 31 ______ d) 53______ e) 2-10______ f) 12300 _____ g) -23_____ h) 5452______ i) 737-5 _______ j) 53 ÷ 52_____ k) 28 ÷2-3______ k) (32)3_______ 1 l) (4-2)3________ m) (83)-5_______ n) -120________ Scientific Notation The number before the “x (multiplication)” is called the coefficient and must be a number between 1 and 10. The number after the “x (multiplication)” is called the base. For example, 1 x 109 coefficient = 1 Standard to Scientific Notation EX 1: 1500 = 1.5 x 103 base = 10 exponent = 9 1.500 {If you make the coefficient 3 powers of 10 smaller (1500 to 1.5) than you must make the exponent 3 powers of 10 higher} Scientific to Standard EX 2: 4 x 105 = 400,000 4.00000 {If you make the exponent 5 powers of 10 smaller (5 to 0) than you must make the coefficient 5 powers of 10 larger (4 to 400,000)} 2) 17,000 = __________ 3) 1.7x 106 = 4) 9 x 10-5 = _____________ 5) 43,800,000 = ____________ _____________ 6) 7 x 1012 = 1) 1000 = _________ _____________ 7) 226,000,000 = ___________ 8) 0.823 =____________________ 9) 0.00467 = _________ Writing numbers in correct SN form 12.53 × 104 = 1.253 ×104+1 = 1.253 ×105 (number gets small; exponents becomes larger) 127.53 × 10-4 = 1.2753 × 10-4 +2 =1.2753 × 10-2 0.0067 × 105 = 6.7 × 105-3 = 6.7 × 102 (number gets big; exponents becomes smaller) 0.012 × 10-3 = 1.2 × 10-3-2 = 1.2 × 10-5 a) 72.35 × 108 = b) 346.026 × 10-5 = ____________ 2 c) 0.0023 × 108 = d) 0.305 × 10-4 =____________ Multiplying Numbers in Scientific Notation When you multiply numbers in scientific notation you multiply the coefficients (the numbers before the “x”) and you add the exponents. (make sure the answer is in scientific notation) 1) (2 x 103) (4 x 106) = 8 x 109 multiply coefficients [2 x 4 = 8] and add exponents [3 + 9 = 12] 2) 5 x 103 (4 x 103) = 20 x 106 = 2.0 x 107 {coefficients; 5 x 4 = 20} & {exponents; 3 + 3 = 6} 3) 2 x 107 (5 x 104) {coefficients; 2 x 5 = 10} & {exponents; 7 + 4 = 11} = 10 x 1011 = 1 x 1012 1) (3 x 103 ) (2 x 104) = ____________ 2) (5.7 x 107) (1.5 x 10-4) = ____________ 3.) (2 .8x 10-9 )(4.75 x 10-3) = ____________ 4) (6 x 10-3 )(4 x 105) = _________________ 5) (7.5 x 105) (5.3 x 102) = _____________ 6) (3.2 x 10-5) )(4.9x 105) = _______________ Dividing Numbers in Scientific Notation When you divide numbers in scientific notation you divide the coefficients (the numbers before the “x”) and you subtract the exponents .(make sure the answer is in scientific notation) Ex.: 6 x 106 = 3 x 102 2 x 104 Ex 8 x 106 = 4 x 10-2 2 x 108 {Divide coefficients [6/2 = 3] and subtract exponents [6 - 4 = 2]} {divide the coefficients [8/2 = 4] and subtract the exponents [6 - 8 = -2]} 1. 8.4 x 109 = __________ 3 x 104 2. 9 x 107 = __________ 4.2 x 105 3. 7 x 108 = __________ 1 x 107 4. 2 x 103 = __________ 3.5 x 103 5. 6 x 104 = __________ 3 x 108 6. 9 x 104 = __________ 3 x 106 3 Adding/Subtracting Numbers in Scientific Notation Step 1 – Rewrite so the exponents are the same Step 2 - add/subtract the decimal Step 3 – Bring down the given exponent on the 10 (make sure the answer is in scientific notation) It is usually easier to change the number with the lowest exponent to the higher exponent to the higher exponent since it represents only a small fraction of the larger number. Ex1: Ex 2: 2 x 103 + 4 x 102 = 2 x 103 + 0.4 x 103 [To raise 4 x 102 one power of 10 to 3 you must reduce the = 2.4 x 103 coefficient by one power of 10] 5 x 107 + 3 x 109 = 0.05 x 109 + 3 x 109 [To raise 5 x 107 two powers of 10 to 9 you must reduce the = 3.05 x 109 coefficient by two powers of 10] 1) 3 x 108 + 5 x 108 = _______________ 2) 6.7 x 107 + 2.4 x 108 = _______________ 3) 8.7 x 108 + 6.3 x 105 = _______________ 4) 4.5 x 107 + 5.2 x 103 = _______________ 5) 8 x 103 + 6 x 105 = _______________ 6) 3.2 x 106 - 2 x 106 = _____________ Ex 1: 4 x 107 - 2 x 106 = 4 x 107 - 0.2 x 107 [To raise 2 x 106 one power of 10 to 7 you must reduce the = 3.8 x 107 coefficient by one power of 10] Ex 2: 8 x 105 - 5 x 103 = 8 x 105 – 0.05 x 105 [To raise 5 x 103 two powers of 10 to 5 you must reduce the = 7.95 x 105 coefficient by two powers of 10] 1) 9 x 108 - 3 x 108 = _______________ 2) 6.8 x 1010 - 4 .9x 108 = _______________ 3) 8 .7x 107 - 2.3 x 104 = _______________ 4) 7 x 104 - 8 x 103 = _______________ 4