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2 ways to write the same number: positive exponents: numbers greater than 1 greater than or equal to one, and less than 10 negative exponents: numbers less than 1, (> 0) (fractions) (always 10) 6,500: standard form 6.5 x 103: scientific notation Express 870,000 in scientific notation 1. Write the number without the comma. 2. Move the decimal point so that the first number will be less than 10 but greater than or equal to 1. 3. Count how many places you had to move the decimal point to get it back to its original spot. This becomes the exponent of 10. Since you moved right and the original number is BIG, the exponent is positive. 4. Drop the zeros after the last nonzero digit. 870000 decimal point is here 870000 x 10 . . 870000 x 10 1 2 3 4 5 8.7 x 105 Express 0.0043 in scientific notation 1. Write the number with the decimal point. 2. Move the decimal so that the first number is 1 or more, but less than 10. 3. Count how many places you had to move the decimal point to get it back to its original spot. The number becomes the exponent of 10. Since you moved left and the original number is SMALL, the exponent is negative. .0043 ? 0043 x 10 . 0043 x 10 . 3 2 1 4.3 x 103 4. Drop the zeros before the nonzero digit. ? Converting to Standard Form Express 3.5 x 104 in standard form 1. Write the coefficient. 3.5 2. Move the decimal point the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. 3.5 4. Fill in with zeros and add commas, as necessary. exponent is 4 00 0 1 2 3 4 35,000 standard form: 4,230 standard form: .0007003 Answer Comparing numbers in scientific notation First compare the exponents. If the exponents are different, the number with the larger exponent is the larger number. 9.99 x 103 < 2.17 x 104 1.02 x 102 > 8.54 x 103 6.83 x 109 < 3.93 x 102 Comparing numbers in scientific notation If the exponents are the same, compare the coefficients. The larger the coefficient, the larger the number. 5.67 x 103 > 4.67 x 103 4.32 x 106 < 4.67 x 106 2.32 x 1010 < 3.23 x 1010 Multiplying Numbers in Scientific Notation 1. Multiply the coefficients 2. Multiply the powers of ten (same base > ADD exponents) 3. Combine the results 4. Put in proper scientific notation form Multiplying Numbers in Scientific Notation Multiply: (6.0 x 104)(2.5 x 102) 1. Multiply the coefficients 6.0 x 2.5 = 15 2. Multiply the powers of ten 104 x 102 = 104 + 2 = 106 3. Combine the results 15 x 106 4. Put in proper form 1.5 x 107 Dividing Numbers in Scientific Notation Dividing with scientific notation follows the same basic rules as multiplying. 1. Divide the coefficients 2. Divide the powers of ten (same base > SUBTRACT exponents) 3. Combine the results 4. Put in proper scientific notation form Division with Scientific Notation Divide: 5.4 x 106 9.0 x 102 1. Divide the coefficients 2. Divide the powers of ten 3. Combine those results 4. Put in proper form 5.4 ÷ 9.0 = 0.6 106 ÷ 102 = 1062 = 104 0.6 x 104 6.0 x 103 Addition and Subtraction with Scientific Notation Numbers in scientific notation can only be added or subtracted if they have the same exponents. If needed, an intermediate step is needed to rewrite one of the numbers so it has the same exponent as the other. Addition and Subtraction (4.0 x 103) + (5.3 x 103 ) Since the exponents are the same (3), just add the coefficients: 4.0 x 103 + 5.3 x 103 = 9.3 x 103 Addition and Subtraction 8.0 x 103 + 5.3 x 103 Since the exponents are the same (3), just add the coefficients. 8.0 x 103 + 5.3 x 103 = 13.3 x 103 This is not proper form, since 13.3 > 10; it should be written as 1.33 x 104 Addition and Subtraction 8.0 x 104 + 5.3 x 103 = Because the exponents are different, convert the number with the smaller exponent to match the other number (Convert 5.3 x 103 to 0.53 x 104) 8.0 x 104 + 0.53 x 10 = 8.53 x 104 4 Since both numbers now have the same exponents, just add the coefficients. Glossary Base The number that is going to be raised to a power. It is multiplied the number of times the power equals. 188 29 73 335 In scientific notation, 112 ten to the power of 3 2154 the base will always = 10 103 = 10 x 10 x 10 = 1,000 Coefficient A number in front of a variable. A factor of a term. 3y 19z 6.5 x 103 scientific notation: .000000459 a coefficient and 10 raised to some power 3.78 x 106 4.59 x 107 Power A number that shows you A quick way to write how many times to use the repeated number in a multiplication. multiplication. scientific notation: a coefficient Exponent some power and 10 raised to or ten to the power of 3 Index 103 = 3.78 x 106 10 x 10 x 10 = 1,000 Blue Whale Scientific Notation A system scientists developed to write big or small numbers using powers of 10 . big numbers 180,000 kg = 1.8 x 105 small numbers a coefficient and 10 raised to some power. 0.00015 kg = 3.78 x 106 1.5 x 104 Standard Form A number whose scientific form has been expanded. 4,500,000 0.00000032 0.006789 120,000 The most "familiar" form of a number. decimal form decimal point Standard Form: 6,500 vs. Scientific Form: 6.5 x 103 6,700,000,000,000 0.000006783 Attachments notebook﴾1801411b2398﴿.galleryitem