1. CONFIDENTIAL 1 Grade 8 Exponents/Scientific Notation 2

2. Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on powers of the base number 10. The second number is called the base. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 10 11 the number 11 is referred to as the exponent or power of ten. Scientific notation The number 123,000,000,000 in scientific notation is written as : 1.23 ×10 11 The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

3. Exponents The "exponent" stands for however many times the thing is being multiplied. The thing that's being multiplied is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power". "5 3 " is "five, raised to the third power". Example 1: 2 x 2 x 2 x 2 x 2 = 2 5 i.e., 2 raised to the fifth power Exponential notation is an easier way to write a number as a product of many factors.

4. Whole numbers can be expressed in standard form, in factor form and in exponential form. Exponential notation makes it easier to write a number as a factor repeatedly. A number written in exponential form is a base raised to an exponent. The exponent tells us how many times the base is used as a factor. Write 10 3, 3 6, and 1 8 in factor form and in standard form. Exponential Form Factor Form Standard Form 10 3 10 × 10 × 101,000 3636 3 ×3 ×3 × 3 × 3 ×3729 1818 1 ×1 × 1 ×1 × 1 × 1 × 1 × 11

5. If a negative number is raised to an even power, the result will be positive. (-2) 4 = - 2 × - 2 × - 2 × - 2 = 16 If a negative number is raised to an odd power, the result will be negative. (-2) 5 = - 2 × - 2 × - 2 × - 2 × - 2 = -32 The negative number must be enclosed by parentheses to have the exponent apply to the negative term. Note that (-2) 4 = - 2 × - 2 × - 2 × - 2 = 16 and -2 4 = -(2 × 2 × 2 × 2) = -16

6. Write in scientific notation: 35,800 Place the decimal point between the first two non-zero numbers, 3 and 5. Since the 3 was in the ten-thousands place, the power of 10 is 10 4. The number in the scientific notation is 3.58 × 10 4. To change the number from scientific notation into standard notation, you can also count the number of times the decimal point moved to determine the power of 10. 35,800 = 3 5 8 0 0 = 3.58 × 10 4 Decimal point moves 4 places to the left.

7. To change the number from scientific notation into standard notation, begin with place value indicated by the power of 10. Add zeroes as place holders when necessary. 0.0079 = 0. 0 0 7 9 = 7.9 × 10 -3 Write in scientific notation: 0.0079 Decimal point moves 3 places to the right.

8. Let us do some practice problems!

9. CONFIDENTIAL 9 1) What is the number 601133 denoted as in the scientific notation? a)60.1133 × 10 5 b)60.1133 × 10 -5 c)6.01133 × 10 5 d)6.01133 × 10- 5 c) 6.01133 × 10 5

10. CONFIDENTIAL 10 2) What is the number 7.56823 × 10 11 denoted as in the standard notation? a)75682300000 b)756823000000 c)7568230000 d)7568230000000 b) 756823000000

11. CONFIDENTIAL 11 3) The speed of sound in air is 331 m/s at 0 o C. Write the speed of sound in air in scientific notation. a)3.31 × 10 5 m/s b)3.31 × 10 4 m/s c)3.31 × 10 3 m/s d)3.31 × 10 2 m/s

12. CONFIDENTIAL 12 4) To write 0.18 ×10- 6 in standard notation, how many times will the decimal point move? a)6 places to the right. b)5 places to the right. c)6 places to the left. d)5 places to the right. c) 6 places to the left.

13. CONFIDENTIAL 13 5) Which number is the smallest? a)1.443 × 10 -9 b)1.443 × 10 -8 c)1.443 × 10 -7 d)1.443 × 10 -11

14. CONFIDENTIAL 14 6) The mean distance of the Earth from the Moon is about 384,400 km. Write the distance in scientific notation. a)3.844 × 10 5 km b)3.844 × 10 4 km c)3.844 × 10 3 km d)3.844 × 10 2 km a) 3.844 × 10 5 km

15. CONFIDENTIAL 15 7) The total area occupied by the state of South Carolina is about 3.2 × 10 5 square miles. Write the number in decimal form. a)32000 square miles b)320000 square miles c)3200 square miles d)3200000 square miles b) 320000 square miles

16. CONFIDENTIAL 16 8) The height of the Empire State building in New York is 1,250 ft. Write this number in scientific notation. a)1.25 × 10- 3 b)1.25 × 10 2 c)1.25 × 10 -2 d)1.25 × 10 3

17. CONFIDENTIAL 17 9)Which year has the highest cost? YearCost 18 × (1.03) 28 × (1.03) 2 38 × (1.03) 3 48 × (1.03) 4 a) 8 × (1.03) 4 b)8 × (1.03) 3 c)8 × (1.03) 2 d)8 × (1.03)

18. CONFIDENTIAL 18 10) The mean distance from Venus to the Sun is about 1.08 × 10 9 km. Write this number in decimal form. a)10800000 b)109000000 c)1080000000 d)109000000 c) 1080000000

19. Assessment

20. CONFIDENTIAL 20 1) What is the number 934000000000 denoted as in the scientific notation? a)5.67× 10 11 b)9.614 × 10 -11 c)9.614 × 10 10 d)9.614 × 10- 10 a) 9.34 × 10 11

21. CONFIDENTIAL 21 2) What is the number 5.67 × 10 8 denoted as in the standard notation? a)56700000 b)567000 c)567000000000 d)567000000

22. CONFIDENTIAL 22 3) A teacher asked four students for the value of 5 × 10 4, Jeff said 5,000, Mike said 50,000, Charles said 500,000 and Nathan said 5,000,000. Who among the four is correct? a)Jeff b)Mike c)Charles d)Nathan b) Mike

23. CONFIDENTIAL 23 4) The total men in the Civilian Labor Force in the U.S in the year 2001 was about 7.5743 × 10 7. Write the number in decimal form. a)75743000000 b)7574300000 c)75743000 d)7574300000000 c) 75743000

24. CONFIDENTIAL 24 5) Which number is the larger than 9.6 × 10 -10 ? a)9.5 × 10 -10 b)10 × 10 -10 c)10 × 10 -11 d)9.6 × 10 -11 b) 10 × 10 -10

25. CONFIDENTIAL 25 6) The total number of farms in the year 2000 in the U.S was about 2,228,000. Write the number in scientific notation. a)2.228 × 10 -6 b)2.228 × 10 -7 c)2.228 × 10 7 d)2.228 × 10 6

26. CONFIDENTIAL 26 7) The number of seconds in a week is 6.048 × 10 8. If this number expressed in scientific notation, the numerical value of the exponent will be: a)604800000 b)604800000000 c)60480000000 d)60480000000000 a) 604800000

27. CONFIDENTIAL 27 8) What is 3×10 4 + 7×10 2 + 7×10 -1 in the standard form? a)30707 b)30700.07 c)30707 d)30700.7

28. CONFIDENTIAL 28 9) A population of bacteria in a laboratory experiment doubles every hour. What will be the population of bacteria after 6 hours, if the initial population of the bacteria is 84? a)84x 2 3 b)84x 2 4 c)84x 2 5 d)84x 2 6

29. CONFIDENTIAL 29 10) There are 20 bears in a zoo. What will be their population after 3 years, if the population doubles each year? a) 20 x 2 3 b)20 x 2 4 c)20 x 2 5 d)20 x 2 6

30. You did a great job today!