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CALCULATIONS USING SCIENTIFIC NOTATION SCIENTIFIC NOTATION AND STANDARD FORM Scientific notation is a way of writing numbers using powers of 10. You write a number in scientific notation using the product of two factors. The first factor is a number greater than one and less than 10 and the second factor is a power of 10. 1, 650, 000, 000 = 1.65i10 9 0.00067 = 6.7i10 !4 The numbers on the left side and not in scientific notation are in standard form. Notice that the exponent shows the number of places that the decimal has moved. Also in scientific notation, numbers greater than one have positive exponents and numbers greater than zero and less than one have negative exponents. Scientific notation is also how our calculators display extremely large numbers. When your calculator shows 1.3 03 it means 1.3 · 103 = 1,300. Example 1 Example 2 Convert to scientific notation. Convert to standard form. a. 17, 000 : Move the decimal between the 1 and 7 (4 places) and since the number was greater than one, the exponent is positive. 17, 000 = 1.7i10 4 1.77i10 !3 : The number is between zero a. and one so move the decimal 3 places to the left. 1.77i10 !3 = 0.00177 b. 0.000000345: Move the decimal between the 3 and 4 (7 places) and since the number was between zero and one, the exponent is negative. 0.000000345 = 3.45i10 !7 2.345i10 6 : The number is greater than b. one so move the decimal 6 places to the right. 2.345i10 6 = 2, 345, 000 Problems Convert each number to scientific notation. 1. 12,500 2. 0.000345 3. 34,000,000 4. 0.007 5. 0.0000368 6. 7 Convert each number to standard form. 7. 3.4i10 7 8. 1.49i10 !2 9. 8i10 !6 10. 7.88i10 9 11. 3.14i10 3 12. 1i10 !1 CALCULATIONS WITH SCIENTIFIC NOTATION You can multiply or divide numbers in scientific notation by using the rules for exponents with the same base. Recall that (if the bases are the same) in a multiplication problem with exponents, the exponents are added and that in a division problem with exponents, the exponents are subtracted. After the computation is completed, you may need to convert the final answer into scientific notation. Examples Compute each product or quotient. Convert the final answer to scientific notation if necessary. a. c. (5.2i10 4 )(6i10 3 ) (5.2)(6)(10 4 )(10 3 ) (31.2)(10 7 ) (3.12i101 )(10 7 ) = 3.12i10 8 b. !6 4 !6 !2 4 ( )( arrange factors multiply change to scientific simplify ( 3.2i10 )( 3i10 ) ( 3.2 ) ( 3) (10 ) (10 ) = ( 9.6 ) (10 ) 3.15 1.4 multiply 10 8 10 3 ) arrange factors = 2.25i10 5 divide d. arrange factors 3.15i10 8 1.4i10 3 5i10 !8 8i10 !5 ( )( 5 8 10 !8 10 !5 ) arrange factors ( 0.625 ) (10 !3 ) divide ( 6.25i10 )(10 ) change to scientific !1 = 6.25i10 !4 !3 simplify Problems Compute each product or quotient. Convert the final answer to scientific notation if necessary. 13. 16. ( 3i10 )( 2i10 ) ( 2.75i10 )( 2.5i10 ) 2 3 !2 8 14. 17. ( 6i10 )( 4i10 ) (1.75i10 )( 8i10 ) 2 5 !2 !8 15. 18. ( 2i10 )( 3.2i10 ) ( 4i10 )( 2.5i10 ) 2 !5 2 !3 19. 8i1012 4i10 7 20. 9i10 5 4i10 7 21. 2.5i10 5 5i10 8 22. 5i10 5 2i10 !8 23. 3i1015 4i10 8 24. 8i10 !5 1.25i10 !8 25. One of the giant pyramids of Egypt is made of about 2.4i10 6 carved blocks of stone. If each block weighs approximately 4.95i10 3 pounds, about how many pounds of block make up the pyramid? 26. The average distance from the sun to the earth is 1.488i1011 meters. If the length of the average paperclip is 3i10 !2 meters, how many paperclips would need to be connected together to reach the sun? Answers 1. 1.25i10 4 2. 3.45i10 !4 3. 3.4i10 7 4. 7i10 !3 5. 3.68i10 !5 6. 7i10 0 7. 34,000,000 8. 0.0149 9. 0.000008 10. 7,880,000,000 11. 3140 12. 0.1 13. 6i10 5 14. 2.4i10 8 15. 6.4i10 !3 16. 6.875i10 6 17. 1.4i10 !9 18. 1i10 0 = 1 19. 2i10 5 20. 2.25i10 !2 21. 5i10 !4 22. 25. 26. 2.5i1013 1.188i1010 pounds 4.96i1012 paperclips 23. 7.5i10 6 24. 6.4i10 3