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David Terr
Ph.D. Math, UC Berkeley

 

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>> P.3. Scientific Notation

 

P.2. Exponents

Exponents are an extremely useful tool in mathematics. In much the same way as multiplication corresponds to repeated addition, exponentiation corresponds to repeated multiplication. For instance, the expression 52 is equal to the number 5*5 or 25, 23 is equal to 2*2*2 or 8, and 106 is equal to 10*10*10*10*10*10 or 1,000,000. In the expression 106, the number 10 is known as the base and the small number 6 to the upper-right of the base is known as the exponent or power. The expression 106 is read "ten raised to the sixth power", or more simply, "ten to the sixth power", or even more concisely, "ten to the sixth". The second power of a number is more commonly known as its square and the third power as its cube.

Exponents may be negative or zero as well as positive. What do we mean by an expression such as 20 or 10-3? Any number raised to the power zero is defined to be equal to one. Thus we have 20 = 1. Furthermore, any number raised to a negative power is defined to be the recipricol of that number raised to the absolute value of that power. Thus, for instance, we have 10-3 = 1/103 = 1/(10*10*10) = 1/1000, or 0.001.

 

Example 1: Evaluate 26.

Solution: We have 26 = 2*2*2*2*2*2 = 64.

 

Example 2: Evaluate 70.

Solution: Since the zeroth power of every number is equal to 1, we have 70 = 1.

 

Example 3: Evaluate 3-4.

Solution: We have 3-4 = 1/34 = 1/(3*3*3*3) = 1/81.

 

Powers of ten are most useful since we count in base 10. Below we list the first several nonnegative powers of 10 as well as their names.

100
1
one
101
10
ten
102
100
one hundred
103
1,000
one thousand
104
10,000
ten thousand
105
100,000
one hundred thousand
106
1,000,000
one million
107
10,000,000
ten million
108
100,000,000
one hundred million
109
1,000,000,000
one billion
1010
10,000,000,000
ten billion
1011
100,000,000,000
one hundred billion
1012
1,000,000,000,000
one trillion

Table P.2.1: Nonnegative Powers of Ten

 

As you can see, these numbers grow very quickly! Thus, positive exponents provide a concise way of expressing very large numbers. In the same way, negative exponents are a convenient way of expressing very small numbers. The following table lists the first several negative powers of ten as well as their names.

10-1
0.1
one tenth
10-2
0.01
one hundredth
10-3
0.001
one thousandth
10-4
0.0001
one ten-thousandth
10-5
0.00001
one hundred-thousandth
10-6
0.000001
one millionth
10-7
0.0000001
one ten-millionth
10-8
0.00000001
one hundred-millionth
10-9
0.000000001
one billionth
10-10
0.0000000001
one ten-billionth
10-11
0.00000000001
one hundred-billionth
10-12
0.000000000001
one trillionth

Table P.2.2: Negative Powers of Ten

 

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