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APES Scientific Notation Handout Scientific notation is based on powers of ten. Numbers that consist of a one followed by several zeros can be written as powers of ten. The power of ten is just the number of zeros in the number. 10 = 101 100 = 102 1,000 = 103 10,000 = 104 100,000 = 105 1,000,000=106 10,000,000 =107 100,000,000 = 108 0.1 = 10-1 0.01 = 10-2 0.001 = 10-3 0.0001 = 10-4 0.00001 = 10-5 0.000001 = 10-6 0.0000001 = 10-7 0.00000001 = 10-8 A large number can be put into scientific notation by writing it as the first digit of a number, followed by a decimal point, followed by all of the other digits in the number, multiplied by a power of 10. The number below is in scientific notation. 8.75 x 108 To convert a number much greater than one to scientific notation, you must do two things: 1. Move the decimal point in the number to the left so that it is between the left-end digit in the number and the digit that is next to the left end. If the number does not have a decimal point to start with, put one on the right-hand end of the number. 2. Multiply the number by a power of ten. The power should be equal to the number of digits that you had to pass when you moved the decimal point to the left. Example 1: Convert the number 8415.7 to scientific notation. Step 1: Move the decimal point between the 8 and the 4. 8.4157 Step 2: Because you moved the decimal point three digits to the left, multiply by ten to the third power. 8.4157 x 103 Example 2: Convert the number 532,000 to scientific notation. Step 1: Since there is no decimal point to start with, place one on the right-hand end of the number. 532,000.0 Step 2: Move the decimal point between the 5 and the 3. 5.32 Step 3: Because you moved the decimal point five digits to the left, multiply by ten to the fifth power. 5.32 x 105 Notice that it is not necessary to keep the extra 0s on the right-hand end of the number. To convert a number much smaller than one to scientific notation, you must do two things: 1. Move the decimal point in the number to the right so that it is just to the right of the first digit that is not zero. 2. Multiply the number by a negative power of ten. The power should be equal to the number of digits that you had to pass when you moved the decimal point to the right. Example 3: Convert the number 0.000657 to scientific notation. Step 1: Move the decimal point between the 6 and the 5. 6.57 Step 2: Because you moved the decimal point four digits to the right, multiply by ten to the negative fourth power. 6. 57 x 10-4 Example 4: Convert the number 0.0000008 to scientific notation. Step 1: Move the decimal point to the right of the eight. 8.0 Step 2: Because you moved the decimal point seven digits to the right, multiply by ten to the negative seventh power. 8.0 x 10-7 Combining Numbers in Scientific Notation Multiplying numbers in scientific notation Think about 102 x 103. The meaning of the exponent 2 in 102 tells you that 102 = 10 x 10 and the meaning of the exponent 3 in 103 tells you that 103 = 10 x 10 x 10. So it must be true that: 102 x 103 = 10 x 10 x 10 x 10 x 10 Therefore, 102 x 103 = 105 To multiply two powers of 10, add the exponents. Example 5: Calculate (2.7 x 108) (5.0 x 106). Step 1: Rearrange the numbers to group the powers of ten together. (2.7 x 5.0) (108 x 106) Step 2: Multiply the numbers in the two sets of parentheses. 13.5 x 108+6 = 13.5 x 1014 Step 3: Rewrite the number in scientific notation. 1.35 x 1015 Example 6: Multiply 3.567 x 10-4 by 8.5 x 10-3 Step 1: Rearrange the numbers to group the powers of ten together. (3.567 x 8.5) (10-4 x 10-3) Step 2: Multiply the numbers in the two sets of parentheses. 30.3195 x 10-4 + (-3) = 30.3195 x 10-7 Step 3: Rewrite the number in scientific notation. 3.03195 x 10-6 Dividing numbers in scientific notation Think about 105/103. This is the same as 10 x 10 x 10 x 10 x 10/10 x 10 x 10 and this fraction can be reduced to 10 x 10. Therefore, 105/103 = 102 To divide one power of 10 by another power of 10, subtract the exponent of the divisor from the exponent of the dividend. Example 7: Divide 2.41 x 102 by 7.4 x 1021 Step 1: Find 2.41/7.4 and subtract the exponents. 0.32568 x 10(2-21) = 0.32568 x 10-19 Adding & Subtracting Numbers in Scientific Notation To add or subtract two numbers in scientific notation: 1. Adjust the powers of 10 in the 2 numbers so that they have the same index. (Tip: It is easier to adjust the smaller index to equal the larger index). 2. Add or subtract the numbers. 3. Give the answer in scientific notation. Example 8: Evaluate 2 × 103 + 3.6 × 104, giving your answer in scientific notation. Example 9: Evaluate 7 × 105 – 5.2 × 104, giving your answer in scientific notation.