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Scientific Notation Remember how? Rules of Scientific Notation 4.23 coefficient 5 x 10 base exponent The coefficient must be greater than or equal to 1 and less than 10. Must be base 10 The exponent shows the number of places the decimal must be moved to change the coefficient to a standard number A standard number exists when the exponent is zero (0) BAD EXAMPLES These are all BAD EXAMPLES of scientific notation. DON’T DO THESE!! Example Why it’s incorrect Corrected 0.34 x 107 Coefficient is not between 1 and 10 3.4 x 106 25 x 10 -5 Coefficient is not between 1 and 10 2.5 x 10-4 4.74 x 28 Not base 10 (we won’t be solving for these) 4.74 x 256 = 1213.44 = 1.21344 x 103 Scientific Notation Standard When going from scientific notation to standard, do the following If the exponent is POSITIVE, move the decimal RIGHT Add place-holder zeroes as needed EX: 3.67 x 105 367000 If the exponent is NEGATIVE, move the decimal LEFT Add place-holder zeroes as needed EX: 7.25 x 10-3 0.00725 Example Write 1.69 x 104 as a standard number 1 6 9 0 0 x 10 41032 Once you get to 100, you’re at the standard number. When recording an answer, DO NOT put the 100. Leave it out. Remember: x100 means x1 Example Write 4.23 x 10-3 as a standard number 0 0 0 4 2 3 x 10 -3-2-10 Once you get to 100, you’re at the standard number. When recording an answer, DO NOT put the 100. Leave it out. Remember: x100 means x1 Also, for neatness, it’s best to include the leading zero before the decimal. Standard Scientific Notation When going from standard to scientific notation, do the opposite as before, so: If you move the decimal LEFT, the exponent is POSITIVE EX: 8976 8.976 x 103 If you move the decimal RIGHT, the exponent is NEGATIVE EX: 0.00058 5.8 x 10-4 Example Write 780374.2 in scientific notation. 7 8 0 3 7 4 2 x 10 7. Is a number between 1 and 10. We needed to move the decimal 5 times to the left, so the exponent became 105. 5 1234 0 Example Write 0.006235 in scientific notation. 0 0 0 6 2 3 5 x 0 -3 -2 1 10 6 is a number between 1 and 10. We needed to move the decimal 3 times to the right, so the exponent became 10-3. Get rid of any leading zeroes. Multiplying in Scientific Notation Example: 3.2 x 104 x 8.7 x 105 Rules: MULTIPLY the coefficients together like usual 3.2 x 8.7 = 27.84 ADD the exponents together 104 x 105 = 109 Readjust for proper scientific notation, if needed 27.84 x 109 2.784 x 1010 Multiplication Practice Problems Problem Work Temp Answer FINAL Answer 4.8 x 103 • 2.3 x 1012 4.8 • 2.3 = 11.04 103 • 1012 = 10(3 + 12) = 1015 11.04 x 1015 Can’t leave 11 1.104 x 1016 3.6 x 10-4 • 2.1 x 103 3.6 • 2.1 = 7.56 10-4 • 103 = 10(-4 + 3)=10-1 7.56 x 10-1 The 7 is ok 7.56 x 10-1 2.65 x 10-5 • 7.3 x 10-7 2.65 • 7.3 = 19.345 10-5 • 10-7 = 10(-5 + -7) = 10-12 19.345 x10-12 Can’t leave 19 1.9345 x 10-11 9.56 x 106 • 9.8 x 10-4 9.56 • 9.8 = 93.688 106 • 10-4 = 10(6 + -4) = 102 93.688 x102 Can’t leave 93 9.3688 x 103 2.1 • 7.22 = 15.162 15.162 x10-16 103 • 10-19 = 10(3 + -19)= 10-16 Can’t leave 15 1.5162 x 10-15 2.1 x 103 • 7.22 x 10-19 Dividing in Scientific Notation Example: 4.76 𝑥 107 8.3 𝑥 103 DIVIDE the coefficients like usual (top divided by bottom) 4.76 8.3 = 0.573 SUBTRACT the exponents (top # – bottom #) 107 103 = 104 Readjust for proper scientific notation, if needed 0.573 x 104 5.73 x 103 Division Practice Problems Problem 3.31 𝑥 103 2.43 𝑥 108 6.7 𝑥 107 8.22 𝑥 103 3.0 𝑥 10−5 7.8 𝑥10−3 4.5 𝑥10−4 2.99 𝑥10−7 4 𝑥107 8.2 𝑥10−9 Work (coeff) 3.31 = 1.36 2.43 6.7 = 0.815 8.22 3.0 = 0.385 7.8 4.5 = 1.51 2.99 4 = 0.488 8.2 Work (exp) 103 = 10−5 8 10 107 = 104 3 10 10−5 −2 = 10 10−3 10−4 = 103 −7 10 107 = 1016 −9 10 Temp Answer FINAL Answer 1.36 x 10-5 1.36 x 10-5 0.815 x 104 8.15 x 103 0.385 x 10-1 3.85 x 10-2 1.51 x 103 1.51 x 103 0.488 x 1016 4.88 x 1015 Scientific Method with Units Metric units have assigned values. When calculating with those values, replace the unit with its value, then solve. The values are NOT the same as the ones for the factor label conversions This is because they are absolute values, not comparisons to the base unit. Unit Value Sample Equivalent (Scientific) Equivalent (Standard) kilo- 103 6.27 kg 6.27 x 103 g 6270 g mega- 106 2.3 MHz 2.3 x 106 Hz 2300 000 Hz nano- 10-9 7.4 nm 7.4 x 10-9 m 0.000 000 007 4 m Practice Problems with Units Problem 24 𝑘𝑔 2𝑔 230 𝑝𝑚 (𝑝𝑖𝑐𝑜) 52 𝑛𝑚 (𝑛𝑎𝑛𝑜) Equivalent 24 𝑥 103 2 Work (coeff) 24 = 12 2 230 = 4.42 52 Work (exp) 103 = 103 0 10 2.3 ks • 16 s 230 𝑥 10−12 52 𝑥10−9 2.3 𝑥 103• 16 10−12 −3 = 10 10−9 2.3 • 16 = 36.8 103 • 100 = 103 0.4 kHz • 98 mHz 0.4 x 103 • 98 x 10-3 0.4 • 98 = 39.2 103 • 10-3 = 100 Answer 12 x 103 1.2 x 104 g 4.42 x 10-3 m (or 4.42 mm) 36.8 x 103 3.68 x 104 39.2 x 100 3.92 x 101 Hz