Negative Exponents and Scientific Notation

Chapter 4.2

x-a =

= xax-a1

1xa

Negative Exponents

a.

x12

1

1. Write with positive exponents.

x-12

b.

w5

1

1. Write with positive exponents.

w-5

z-2

z2

1

1. Write with positive exponents.

c.

a.

43

1

641

2. Evaluate.

4-3

b.

24

1

161

2. Evaluate.

2-4

w4

a.

3

w-4

3

3. Simplify. Write the expression with no negative exponents.

b.

x6

z-2

x-6

y4z2

3. Simplify. Write the expression with no negative exponents.

y4

c.

x6y5

1

3. Simplify. Write the expression with no negative exponents.

x-6y-5

a. (

(2)-2

4x8

y10

x-8 y10

4. Write with positive exponents.

2x4y-5) -2

Use the power rules.Evaluate and write with positive exponents.

b.

y2

y2

y-3

y3z4

z6

y5

z2

4. Write with positive exponents.

z-4

z-6

Write with positive exponents.Use the quotient rule.Use the product rule.

x-a =

= xax-a1

1xa

Negative Exponents

Scientific Notation

a

a is a number so that 1 ≤ a < 10

n is an integer

x 10n

Two methods to change a number into Scientific Notation

1. Move the decimal point left or right. The number of places is n (left is positive, right is negative)

a x 10n

2. n represents the place value of the first nonzero digit from the original number.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

Here are the place names.These are the exponential form for each place.

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

x 104

a. 78,200

78200.

7.82

5. Write in scientific notation.

Count the number of places to move.Positive 4 is the exponent.

x 104a. = 7.82

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

5. Write in scientific notation.

8,2007

What is the place value of 7?

b. 4,786,000

4786000.

x 1064.786

5. Write in scientific notation.

Count the number of places to move.Positive 6 is the exponent.

5. Write in scientific notation.

x 106b. = 4.786

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

,786,0604

What is the place value of4?

a. 0.98

098.

x 10-19.8

6. Write in scientific notation.

Count the number of places to move.Negative 1 is the exponent.

6. Write in scientific notation.

x 10-1a. = 9.8

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

0. 89

What is the place value of 9?

b. 0.000092

0000092.

x 10-59.2

6. Write in scientific notation.

Count the number of places to move.Negative 5 is the exponent.

6. Write in scientific notation.

x 10-5b. = 9.2

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

0.0000 29

What is the place value of 9?

a. 1.93 x 10

193.

1,930,000

0000

7. Write in decimal notation.

The exponent is positive 6.Move the decimal point right 6 places.

6

a. 1.93 x = 1,930,000

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

7. Write in decimal notation.

106

What is the place value of 1?

b. 8.562 x 10

8562.

0.00008562

00000

7. Write in decimal notation.

The exponent is negative 5.Move the decimal point left 5 places.

-5

7. Write in decimal notation.

b. 8.562 x = 0.00008562

billions

hundred millions

ten millions

millions

hundred thousands

ten thousands

thousands

hundreds

tens

ones

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

millionths

.

109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

10-5

What is the place value of 8?

8. Astronomers measure distances to faraway galaxies in parsecs. A parsec is 30,900,000,000,000,000 meters, Write this in scientific notation.

30900000000000000.

3.09 x 1016 m

Using Scientific Notation

a x 10n

a is a number so that 1 ≤ a < 10

n is an integer

)1.4

9. Use scientific notation and the laws of exponents to find the following. Leave your answer in scientific notation.

a. (

(

7.84 x 1013

(

56,000)(1,400,000,000)

5.6 x 104) x 109

Write each in scientific notation.

Use the product rule.Multiply the numbers.

b.

9. Use scientific notation and the laws of exponents to find the following. Leave your answer in scientific notation.

3.7

0.3 x

0.0001110.00000037

1.11 x 10-4

x 10-7

Write each in scientific notation.

Write as positive exponents.Divide the numbers.

=

107

104

b.

9. Use scientific notation and the laws of exponents to find the following. Leave your answer in scientific notation.

3.7

0.3 x

0.0001110.00000037

1.11 x 10-4

x 10-7

Write in scientific notation.

Use the product rule.Use the quotient rule.

=

107

104

3 x 10-1 x 103

3 x 102

1013

10. The average from Earth to the distant star Betelgeuse . would it take a space probe to travel from Earth to Betelgeuse at a ? Round to three significant digits.

=(3.09 x 1013)

t = ds

(159)(50000)

((

distanceis 159 parsecs

(1 parsec = 3.09 x 1013 km)speed of 50,000 kilometers per hour

How many hours

Write in scientific notation.

Multiply and use the product rule.

1.59 x 102) 3.09 x )5 x 104

4.9131x 1015

5 x 104

0.98262=

9.83

x

≈

9.8262 x

x 1010

Divide and use the quotient rule.

Write in scientific notation.

4.9131x 1015

5 x 104

Round to three significant digits.

1011

x 10-1

Use the product rule.

1011

Using Scientific Notation

a x 10n

a is a number so that 1 ≤ a < 10

n is an integer

Scientific Notation

a x 10n

a is a number so that 1 ≤ a < 10

n is an integer

Negative Exponents and Scientific Notation

Chapter 4.2