Scientific -Notation

Scientists need to express small measurements, such as themass of the proton at the center of a hydrogen atom(0.000 000 000 000 000 000 000 000 001 673 kg), and largemeasurements, such as the temperature at the center ofthe Sun (15 000 000 K). To do this conveniently, theyexpress the numerical values of small and largemeasurements in scientific notation, which has two parts.

An exponentof 10 by whichthe number ismultiplied

A number in whichonly one digit isplaced to the leftof the decimal

Thus, the temperature of the Sun, 15 million kelvins, iswritten as 1.5 x 107 K in scientific notation.

Positive Exponents Express 1234.56 in scientific notation.

1234.56

1234.56 x 10° = 123.456 X 101v123.456 X 101 = 12.3456 X 102

v12.34S6 X 102 = 1.23456 x 103v

1.234 56 x 103

Each timethe decimalplace ismoved oneplace to theleft,

theexponent isincreased byone.

Negative Exponents Express 0.006 57 in scientific notation.

0.006157

0.006 57 x 10° = 0.0657 X 10-1V

0.0657 X 10-1 = 0.657 X 10-2v

0.657 X 10-2 = 6.57 X 10-3v6.57 X 10-3

Each timethe decimalplace ismoved oneplace to theright,

theexponent isdecreasedby one.

SCIENTIFIC NOTATION Name _

Scientists very otten deal with very small and very large numbers, which can lead to a ~fconfusion when counting zeros! We have learned to express these numbers as powers vi' 1D.

ScienHfic notation takes the form of M x 10n where 1 sM < 10 and \\nH represents 'the numberof decimal places to be moved. Posltlve n Indicates the standard form Isa large number.Negative n Indicates a number between zero and one,

Example 1: Convert 1,500,000to scientific notation.We move the decimal point so that there is onlyone digit to its left, a total of 6 places.

1.500,000 = 1.5 x 106I!xample 2: Convert 0.000025 to scientific notation.F:orthis, we move the decimal point 5 places to theright.

0.000025 = 2.5 x 1()-5(Note that when a number starts out less 'than one,the exponent is always negative.)

Convert the following to scientific notation.

1. 0.005 = 6. 0.25 =2. 5,050 = 7. 0.025 =

3. 0.0008 = 8. 0.0025 =4. 1,000 = 9. 500=

5. 1,000,000 = 10. 5,000 =

Convert the following to standard notcnon,

l. 1.5 x 1()3 = 6. 3.35, X 10-1 =

2. 1.5 X 10-3 = 7. 1.2 X 10-4 =3. 3.75 X 10-2 = 8. 1 x 104 =4. 3.75 x 1()2 = 9. 1 X 10-1 =5. 2.2 x lOS = 10, 4 x 100 =

1. Express each of the following numbers in scientific notation.

a. 230

b. 5601

c. 14 100 000

d. 56 million

e. 2/10

f. 0.45013

g. 0.089

h. 0.00026

i. 0.000 000 698

j. 12 thousandth

2. Express each of the following measurements in scientific notation.a. speed of light in a vacuum, 299 792 458 m/s

b. number of seconds in a day, 86 400 s

c. mean radius of Earth, 6378 km

d. density of oxygen gas at oDe and pressure of 101 kPa, 0.001 42 g/mL

e. radius of an argon atom, 0.000 000 000 098 m

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Operatilans with, Sc~i:el1litificNotation Operations withScientific Notation1. Perform the following operations and express the answers in scientific notation.

a. 0.2 X 105) + (5.35 X 106)

b. (6.91 X 10-2) + (2.4 X 10-3)

e. (8.41 x 10-5) - (7.9 x 10-6)

2. Perform the following operations and express the answers in scientific notation.a. (4.3 X 108) x (2.0 x 106)

b. (6.0 X 103) x (1.5 x 10-2)

c. (1.5 X 10-2) x (8.0 x 10-1)

7.8 X 103d. 1.2 X 104

8.1 X 10-2e. 9.0 X 102

6.48 X 105