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Name: _________________________________________________________ Period: ______ Date: ___________________________ 11th Grade Mathematics PSSA Preparation Program Mastered On: _____________________ Scientific Notation Anchors Addressed M11.A.1.1.2 – Express numbers and/or simplify expressions using scientific notation (including numbers less than 1.) Concepts Scientific Notation is a method in which numbers, regardless of how large or small, can be efficiently written using powers of ten. Numbers written in scientific notation consist of two factors: 10 , where a is a number between 1 and 10, but not 10, and n is the power of ten. Converting Numbers into Scientific Notation STEP 1: Move the decimal place so that it is to the right of the first non-zero digit. (So the value of the number is between 1 and 10. The number can be 1 but not 10. STEP 2: Count the number of places (n) the decimal point moved and which direction. STEP 3: If the decimal point moved to the left, write as: 10 . If the decimal moved to the right, write as: 10 . Example 1: Write 12,723,923 in scientific notation. Solution: Move the decimal point to the left until the number is between 1 and 10 as shown: 1.2723923 Now, count the number of places the decimal point moved. This number (7) is your exponent, therefore 12,723,923 written in scientific notation is: 1.2723923 10 Example 2: Write 0.00473562 in scientific notation. Solution: Move the decimal point to the right until the number is between 1 and 10 as shown: 4.73562 Now, count the number of places the decimal point moved. This number is your exponent, therefore 0.00473562 written in scientific notation is: 4.73562 10 Calculator Tip: Most scientific calculators will convert number to and from scientific notation. Many calculators use EE to represent 10 . Therefore, if you wanted to enter 2.3 10 into the calculator, type /80%C1. Converting a Number from Scientific Notation STEP 1: Write the value of a. STEP 2: If n is negative, add − 1 zeros and a decimal point before the value of a. If n is positive, count the number of places after the decimal point in a, then add zeros until the number of places after the decimal point is n. Finally, erase the decimal point from the middle of a from STEP 1. Example 3: Write 3.14 10 Solution: First write the value of a: 3.14. Since n is positive and there are two digits after the decimal places, add three zeros to the end and remove the decimal point from between the 3 and 1, therefore: 3.14 10 is 314,000. Example 4: Write 2.4 10 Solution: First write the value of a: 2.4. Since n is −4, add three zeros before the 2 and move the decimal point to the left of the zeros. Therefore, 2.4 10 is 0.00024. Exercises A. Convert the following numbers into scientific notation. 1. 7,475,000 3. 0.0002458 5. 1,010 2. 0.0547 4. 302 6. 0.000014 The following exercises work the same way as above, but may look different. Some numbers are written in a manner similar to scientific notation, but need adjusted because the value of a is not between 1 and 10. In these cases, if you make the value of a smaller, then you need to make the value of n larger. Conversely, if you make the value of a larger, then you need to make the value of n smaller. Consider the example below: Example 5: Convert 42.5 10 into scientific notation. Solution: Since 42.5 is larger than 10, moving the decimal point one place to the left makes the value of a smaller by one place. Therefore, increasing n by one place yields: 4.25 10 . 9. 42 11. 1.4 13. 37.8 10 10. 0.58 10 12. 356.2 10 14. 56.24 10 B. Convert the following numbers to standard notation. 15. 3.02 10 17. 7.6 10 19. 1.2574 10 16. 5.4 10 18. 8.56 10 20. 5.4 10 C. Answer the following questions. 21. On August 25, 2011, the U.S. national debt was $14,648,835,000,000, of which $3,654,861,000 was the total interest paid to lenders in 2011. Write the national debt and the interest paid in 2011 in scientific notation. 22. Using the debt figure from question 21, if each citizen were to pay his/her share of the debt, how many citizens are there in the United States if each person is responsible for $46,941? 23. Many monuments in Washington D.C. were built out of marble, a very heavy stone mined in the United States. Convert the following statistics about monuments in standard notation. A. Lincoln Memorial weighs 7.6 10 pounds. B. Tomb of the Unknown Soldier weighs 1.58 10 pounds. C. Washington Monument weighs 9.0854 10 tons and was built using marble, granite, and limestone. 24. Compare the weights of the monuments from question 23. List the weights from least to greatest on the lines below. (Recall that 1 ton is 2,000 pounds.) __________________________________________________________________ __________________________________________________________________ __________________________________________________________________