Scientific NotationRemember how?

The coefficient must be greater than or equal to 1 and less than 10.

Must be base 10 The exponent shows the number of places the

decimal must be moved to change the coefficient to a standard number

A standard number exists when the exponent is zero (0)

Rules of Scientific Notation

4.23 x 105

coefficient base exponent

These are all BAD EXAMPLES of scientific notation. DON’T DO THESE!!

BAD EXAMPLES

Example Why it’s incorrect

Corrected

0.34 x 107 Coefficient is not between 1 and 10

3.4 x 106

25 x 10 -5 Coefficient is not between 1 and 10

2.5 x 10-4

4.74 x 28 Not base 10(we won’t be solving for these)

4.74 x 256 = 1213.44 = 1.21344 x 103

When going from scientific notation to standard, do the following If the exponent is POSITIVE, move the decimal

RIGHT Add place-holder zeroes as needed EX: 3.67 x 105 367000

If the exponent is NEGATIVE, move the decimal LEFT Add place-holder zeroes as needed EX: 7.25 x 10-3 0.00725

Scientific Notation Standard

Write 1.69 x 104 as a standard number

Example

1 6 9 0 0 x 10 4

3210

Once you get to 100, you’re at the standard number. When recording an answer, DO NOT put the 100. Leave it out. Remember: x100 means x1

Write 4.23 x 10-3 as a standard number

Example

0 0 0 4 2 3 x 10 -3

-2-10

Once you get to 100, you’re at the standard number. When recording an answer, DO NOT put the 100. Leave it out. Remember: x100 means x1Also, for neatness, it’s best to include the leading zero before the decimal.

When going from standard to scientific notation, do the opposite as before, so: If you move the decimal LEFT, the exponent is

POSITIVE EX: 8976 8.976 x 103

If you move the decimal RIGHT, the exponent is NEGATIVE EX: 0.00058 5.8 x 10-4

Standard Scientific Notation

Write 780374.2 in scientific notation.

Example

7 8 0 3 7 4 2 x 10

012345

7. Is a number between 1 and 10. We needed to move the decimal 5 times to the left, so the exponent became 105.

Write 0.006235 in scientific notation.

Example

0 0 0 6 2 3 5 x 10

0-1-2-3

6 is a number between 1 and 10. We needed to move the decimal 3 times to the right, so the exponent became 10-3. Get rid of any leading zeroes.

Example: 3.2 x 104 x 8.7 x 105

Rules: MULTIPLY the coefficients together like usual

3.2 x 8.7 = 27.84 ADD the exponents together

104 x 105 = 109

Readjust for proper scientific notation, if needed 27.84 x 109 2.784 x 1010

Multiplying in Scientific Notation

Multiplication Practice Problems

Problem Work Temp Answer

FINAL Answer

4.8 x 103 • 2.3 x 1012

4.8 • 2.3 = 11.04103 • 1012 = 10(3 + 12) = 1015

11.04 x 1015

Can’t leave 11

1.104 x 1016

3.6 x 10-4 • 2.1 x 103

3.6 • 2.1 = 7.5610-4 • 103 = 10(-4 +

3)=10-1

7.56 x 10-1

The 7 is ok7.56 x 10-1

2.65 x 10-5 • 7.3 x 10-7

2.65 • 7.3 = 19.34510-5 • 10-7 = 10(-5 + -7) = 10-12

19.345 x10-12

Can’t leave 19

1.9345 x 10-

11

9.56 x 106 • 9.8 x 10-4

9.56 • 9.8 = 93.688106 • 10-4 = 10(6 + -4) = 102

93.688 x102

Can’t leave 93

9.3688 x 103

2.1 x 103 • 7.22 x 10-19

2.1 • 7.22 = 15.162103 • 10-19 = 10(3 + -

19)= 10-16

15.162 x10-16

Can’t leave 15

1.5162 x 10-

15

Example: DIVIDE the coefficients like usual (top divided by

bottom)

SUBTRACT the exponents (top # – bottom #)

Readjust for proper scientific notation, if needed 0.573 x 104 5.73 x 103

Dividing in Scientific Notation

Division Practice Problems

Problem Work (coeff)

Work (exp)

Temp Answer

FINAL Answer

1.36 x 10-5 1.36 x 10-5

0.815 x 104 8.15 x 103

0.385 x 10-1 3.85 x 10-2

1.51 x 103 1.51 x 103

0.488 x 1016 4.88 x 1015

Metric units have assigned values. When calculating with those values, replace the unit with its value, then solve.

The values are NOT the same as the ones for the factor label conversions This is because they are absolute values, not

comparisons to the base unit.

Scientific Method with Units

Unit Value Sample Equivalent (Scientific)

Equivalent (Standard)

kilo- 103 6.27 kg 6.27 x 103 g 6270 g

mega- 106 2.3 MHz 2.3 x 106 Hz 2300 000 Hz

nano- 10-9 7.4 nm 7.4 x 10-9 m 0.000 000 007 4 m

Practice Problems with Units

Problem Equivalent Work (coeff)

Work (exp) Answer

12 x 103 1.2 x 104 g

4.42 x 10-3

m(or 4.42 mm)

2.3 ks • 16 s 16 2.3 • 16 = 36.8

103 • 100 = 103

36.8 x 103 3.68 x 104

0.4 kHz • 98 mHz

0.4 x 103 • 98 x 10-3

0.4 • 98 = 39.2

103 • 10-3 = 100

39.2 x 100 3.92 x 101 Hz