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Chapter 5—Powers, Exponents, and Roots

5-6 Powers-of-10 and Scientific Notation

Our decimal number system was originally invented to deal with everyday situations taking care of business and building things. As a result, the numbers that we encounter in everyday situations aren't particularly large numbers nor are they particularly small. Things have changed a lot over the centuries, however, and there is a need for working with some very large numbers (the size of the universe) and very small numbers (the mass of an atomic sub-particle). Today's versions of engineering, science, medical technology, business, and computer science use scientific notation to express the size, mass, and amounts of all sorts of things. Scientific notation makes it possible and certainly a lot easier to express values and do computations with very small, very ordinary, and very large quantities.

Scientific notation is one of two standards for showing power-of-10 values. The purpose of these standards is to allow the expression of  very large and very small numbers without using huge numbers of digits and decimal places.  It's all done with powers of ten. Here are some examples that demonstrate this fact:

  • A very large number such as 2,000,000,000 can be written with scientific notation as 2 x 109
  • A very small number such as 0.000000674 can be written with scientific notation as 6.74 x 10-7

Expressions in shown in scientific notation are much simpler than their full decimal equivalents. The abbreviated notations are easier to read, understand, write, and work with.

What does an expression such as  2 x 109 mean? From your previous lessons, you should know:

2 x 109 = 2 x 1,000,000,000

An what about  6.74 x 10-7?  That is the same as  6.74 x 10-7 or:

6.74 x 10-7 = 0.000000674

Definition

The main parts of scientific notation are:

  • The coefficient —  the decimal part
  • The base always 10 for scientific notation
  • The exponent for the base  10

Rule

Numbers are written in proper scientific when there is exactly one digit (not including a zero) to the left of the decimal point. This sometimes called the normalized form.

Examples

Are the the following  values expressed in proper (normalized) scientific notation? Explain your response.

2.4 x 102 Yes, because there is exactly one non-zero digit to the left of the decimal point in the coefficient.
67.48 x 104 No, because there is more than one digit to the left of the decimal point in the coefficient.
4.5 x 106 Yes, because there is exactly one non-zero digit to the left of the decimal point in the coefficient.
0.25 x 103 No, because the digit to the left of the decimal point  in the coefficient is a zero.

Rewriting Decimal Values in Scientific Notation

Rewriting any decimal value with scientific notation is a matter of setting the decimal point where there is just one non-zero digit to the right of that point.

The first step in normalizing any value for scientific notation is to determine where the decimal point belongs. That, of course, is the position that leaves just one, non-zero digit to the left of that decimal point.

Examples

For each of the integer values shown below, rewrite them with the decimal  point in the position required for normalized scientific notation:

1.  827= _____  

Ans:  8.27

2.  652000 = _____
Ans:  6.52
3.  0.376 = _____

Ans:  3.76

4.  -55.78 = _____
Ans:  -5.578
6.  500.032 = _____

Ans:  5.00032

6.  0.000045 = _____

Ans:  4.5

The second step is to multiply the result of the first step by a power-of-ten that restores its original  value.

Examples

Original Expression Placement of
Decimal  Point
Multiplier
827    8.27 100 or 102
652000 6.52 105
0.376 3.76 0.1 or 10-1
-55.78 -5.578 10 or 101
500.032 5.00032 102
0.000045 4.5 10-5

 

The final step is to show the value in normalized scientific notation.

Examples

Original Expression Placement of
Decimal  Point
Multiplier Normalized
Scientific Notation
827    8.27 100 or 102  8.27 x 102
652000 6.52 105 6.52 x 105
0.376 3.76 0.1 or 10-1 3.76 x 10-1
-55.78 -5.578 10 or 101 -5.578 x 101
500.032 5.00032 102 5.00032 x 102
0.000045 4.5 10-5 4.5 x 10-5

 

Procedure

To rewrite any decimal value to normalized scientific notation:

  1. Place the decimal point in the coefficient where there is a single non-zero digit to the left of the point.
  2. Determine the amount the original value must be multiplied or divided to make the the decimal placement
  3. Express the amount of multiplication/division in power-of-10 notation.

 

Examples & Exercises

Rewriting Decimal Values in Scientific Notation

Given a value expressed in decimal form, rewrite it with scientific notation. Continue working these exercises until you can do them consistently without error.

 

Now consider a very common situation where the original value is already expressed with some level of power-of-10 notation, but needs to  be adjusted for normalized  scientific notation. Here are a few examples:

Examples

The first step in normalizing any value for scientific notation is to determine where the decimal point belongs. That, of course, is the position that leaves just one, non-zero digit to the left of that decimal point.

Original Expression Placement of
Decimal  Point
22 x 104 2.2
34.6 x 1016 3.4
0.004 x 10-8 4.0
505 x 103 5.05
128000 x 10 -16 1.28
321.55 x 10-2 3.2155

 

Examples

The second step is to multiply the result of the first step by a power-of-ten that would restore its original  value of the coefficient.

Original Expression Placement of
Decimal  Point
Multiplier
22 x 104 2.2 10 or 101
34.6 x 1016 3.4 10 or 101
0.004 x 10-8 4.0 10-3
505 x 103 5.05 102
128000 x 10 -16 1.28 105
321.55 x 10-2 3.2155 102

 

Examples

The third step is to combine the original power-of-ten expression with the one you just created for adjusting the decimal point. Multiply them.

 

Original Expression Placement of
Decimal  Point
Multiplier Combining
10s Portions
22 x 104 2.2 10 or 101 104 x 101 = 105
34.6 x 1016 3.4 10 or 101 1016 x 101 = 1017
0.004 x 10-8 4.0 10-3 10-8 x 10-3 = 10-11
505 x 103 5.05 102 103 x 102 = 105
128000 x 10 -16 1.28 105 10 -16 x 105 = 10-11
321.55 x 10-2 3.2155 102 10-2 x 102 = 1

 

Examples

The final step is to complete the normalized version with the adjusted coefficient and combined powers of ten.

 

Original Expression Placement of
Decimal  Point
Multiplier Combining
10s Portions
Normalized
Scientific Notation
22 x 104 2.2 10 or 101 104 x 101 = 105 2.2 x 105
34.6 x 1016 3.4 10 or 101 1016 x 101 = 1017 3.4 x 1017
0.004 x 10-8 4.0 10-3 10-8 x 10-3 = 10-11 4.0 x 10-11
505 x 103 5.05 102 103 x 102 = 105 5.05 x 105
128000 x 10 -16 1.28 105 10 -16 x 105 = 10-11 1.28 x 10-11
321.55 x 10-2 3.2155 102 10-2 x 102 = 1 3.2155

 

Examples

Rewrite the following decimal values in the normalized scientific form (one, and only one digit, on the left side of the decimal point).

Examples & Exercises

Normalizing for Scientific Notation

Express the given term in scientific form. Round your answers to one decimal place.

 

 

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015