top
Chapter 5—Powers, Exponents, and Roots
56 Powersof10 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 subparticle). 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 powerof10 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 10^{9}
 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 10^{9} mean? From your previous lessons, you should know:
2 x 10^{9} = 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 10^{2}  Yes, because there is exactly one nonzero digit to the left of the decimal point in the coefficient. 
67.48 x 10^{4}  No, because there is more than one digit to the left of the decimal point in the coefficient. 
– 4.5 x 10^{6}  Yes, because there is exactly one nonzero digit to the left of the decimal point in the coefficient. 
0.25 x 10^{3}  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 nonzero 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, nonzero 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 poweroften that restores its original value.
Examples
Original Expression  Placement of Decimal Point  Multiplier 
827  8.27  100 or 10^{2} 
652000  6.52  10^{5} 
0.376  3.76  0.1 or 10^{1} 
55.78  5.578  10 or 10^{1} 
500.032  5.00032  10^{2} 
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 10^{2}  8.27 x 10^{2} 
652000  6.52  10^{5}  6.52 x 10^{5} 
0.376  3.76  0.1 or 10^{1}  3.76 x 10^{1} 
55.78  5.578  10 or 10^{1}  5.578 x 10^{1} 
500.032  5.00032  10^{2}  5.00032 x 10^{2} 
0.000045  4.5  10^{5}  4.5 x 10^{5} 
Procedure To rewrite any decimal value to normalized scientific notation:  Place the decimal point in the coefficient where there is a single nonzero digit to the left of the point.
 Determine the amount the original value must be multiplied or divided to make the the decimal placement
 Express the amount of multiplication/division in powerof10 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 powerof10 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, nonzero digit to the left of that decimal point.
Original Expression  Placement of Decimal Point 
22 x 10^{4}  2.2 
34.6 x 10^{16}  3.4 
0.004 x 10^{8}  4.0 
505 x 10^{3}  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 poweroften that would restore its original value of the coefficient.
Original Expression  Placement of Decimal Point  Multiplier 
22 x 10^{4}  2.2  10 or 10^{1} 
34.6 x 10^{16}  3.4  10 or 10^{1} 
0.004 x 10^{8}  4.0  10^{3} 
505 x 10^{3}  5.05  10^{2} 
128000 x 10^{ 16}  1.28  10^{5} 
321.55 x 10^{2}  3.2155  10^{2} 
Examples
The third step is to combine the original poweroften 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 10^{4}  2.2  10 or 10^{1}  10^{4 }x 10^{1} = 10^{5} 
34.6 x 10^{16}  3.4  10 or 10^{1}  10^{16 }x 10^{1} = 10^{17} 
0.004 x 10^{8}  4.0  10^{3}  10^{8 }x 10^{3} = 10^{11} 
505 x 10^{3}  5.05  10^{2}  10^{3 }x 10^{2} = 10^{5} 
128000 x 10^{ 16}  1.28  10^{5}  10^{ 16 }x 10^{5} = 10^{11} 
321.55 x 10^{2}  3.2155  10^{2}  10^{2 }x 10^{2} = 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 10^{4}  2.2  10 or 10^{1}  10^{4 }x 10^{1} = 10^{5}  2.2 x 10^{5} 
34.6 x 10^{16}  3.4  10 or 10^{1}  10^{16 }x 10^{1} = 10^{17}  3.4 x 10^{17} 
0.004 x 10^{8}  4.0  10^{3}  10^{8 }x 10^{3} = 10^{11}  4.0 x 10^{11} 
505 x 10^{3}  5.05  10^{2}  10^{3 }x 10^{2} = 10^{5}  5.05 x 10^{5} 
128000 x 10^{ 16}  1.28  10^{5}  10^{ 16 }x 10^{5} = 10^{11}  1.28 x 10^{11} 
321.55 x 10^{2}  3.2155  10^{2}  10^{2 }x 10^{2} = 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.  