CONFIDENTIAL 1

Grade 8 Grade 8

Exponents/Scientific Exponents/Scientific Notation 2Notation 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 1011 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 ×1011

The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

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". "53" is "five, raised to the third power".

Example 1: 2 x 2 x 2 x 2 x 2 = 25

i.e., 2 raised to the fifth power

Exponential notation is an easier way to write a number as a product of many factors.

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 103, 36, and 18 in factor form and in standard form.

ExponentialForm

FactorForm

StandardForm

103 10 × 10 × 10 1,00036 3 ×3 ×3 × 3 × 3 ×3 72918 1 ×1 × 1 ×1 × 1 × 1 × 1 × 1 1

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 -24 = -(2 × 2 × 2 × 2) = -16

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 104.

The number in the scientific notation is 3.58 × 104.

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 × 104

Decimal point moves 4 places to the left.

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.

Let us do some practice problems!

CONFIDENTIAL 9

1) What is the number 601133 denoted as in the scientific notation?

a) 60.1133 × 105

b) 60.1133 × 10-5

c) 6.01133 × 105

d) 6.01133 × 10-5

c) 6.01133 × 105

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2) What is the number 7.56823 × 1011 denoted as in the standard notation?

a) 75682300000b) 756823000000c) 7568230000d) 7568230000000

b) 756823000000

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3) The speed of sound in air is 331 m/s at 0oC. Write the speed of sound in air in scientific notation.

a) 3.31 × 105 m/sb) 3.31 × 104 m/sc) 3.31 × 103 m/sd) 3.31 × 102 m/s

d) 3.31 × 102 m/s

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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.

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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

d) 1.443 × 10-11

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6) The mean distance of the Earth from the Moon is about 384,400 km. Write the distance in scientific notation.

a) 3.844 × 105 kmb) 3.844 × 104 kmc) 3.844 × 103 kmd) 3.844 × 102 km

a) 3.844 × 105 km

CONFIDENTIAL 15

7) The total area occupied by the state of South Carolina is about 3.2 × 105 square miles. Write the

number in decimal form.

a) 32000 square milesb) 320000 square milesc) 3200 square milesd) 3200000 square miles

b) 320000 square miles

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 × 102

c) 1.25 × 10-2

d) 1.25 × 103

d) 1.25 × 103

CONFIDENTIAL 17

9)Which year has the highest cost?

Year Cost1 8 × (1.03)2 8 × (1.03)2

3 8 × (1.03)3

4 8 × (1.03)4

a) 8 × (1.03)4

a) 8 × (1.03)4

b) 8 × (1.03)3

c) 8 × (1.03)2

d) 8 × (1.03)

CONFIDENTIAL 18

10) The mean distance from Venus to the Sun is about 1.08 × 109 km. Write this number in decimal

form.

a) 10800000b) 109000000c) 1080000000d) 109000000

c) 1080000000

Assessment

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1) What is the number 934000000000 denoted as in the scientific notation?

a) 5.67× 1011

b) 9.614 × 10-11

c) 9.614 × 1010

d) 9.614 × 10-10

a) 9.34 × 1011

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2) What is the number 5.67 × 108 denoted as in the standard notation?

a) 56700000b) 567000c) 567000000000d) 567000000

d) 567000000

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3) A teacher asked four students for the value of 5 × 104, 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) Jeffb) Mikec) Charlesd) Nathan

b) Mike

CONFIDENTIAL 23

4) The total men in the Civilian Labor Force in the U.S in the year 2001 was about 7.5743 × 107. Write the

number in decimal form.

a) 75743000000b) 7574300000c) 75743000d) 7574300000000

c) 75743000

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

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 × 107

d) 2.228 × 106

d) 2.228 × 106

CONFIDENTIAL 26

7) The number of seconds in a week is 6.048 × 108. If this number expressed in scientific notation, the

numerical value of the exponent will be:

a) 604800000b) 604800000000c) 60480000000d) 60480000000000

a) 604800000

CONFIDENTIAL 27

8) What is 3×104 + 7×102 + 7×10-1 in the standard form?

a) 30707 b) 30700.07c) 30707d) 30700.7

d) 30700.7

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 23

b) 84x 24

c) 84x 25

d) 84x 26

d) 84x 26

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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 23

a) 20 x 23

b) 20 x 24

c) 20 x 25

d) 20 x 26

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