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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 5^{2} is equal to the number 5*5 or 25, 2^{3} is equal to 2*2*2 or 8, and 10^{6} is equal to 10*10*10*10*10*10 or 1,000,000. In the expression 10^{6}, the number 10 is known as the base and the small number 6 to the upperright of the base is known as the exponent or power. The expression 10^{6} 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 2^{0} or 10^{3}? Any number raised to the power zero is defined to be equal to one. Thus we have 2^{0} = 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} = 1/10^{3} = 1/(10*10*10) = 1/1000, or 0.001.
Example 1: Evaluate 2^{6}.
Solution: We have 2^{6} = 2*2*2*2*2*2 = 64.
Example 2: Evaluate 7^{0}.
Solution: Since the zeroth power of every number is equal to 1, we have 7^{0} = 1.
Example 3: Evaluate 3^{4}.
Solution: We have 3^{4} = 1/3^{4} = 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.
10^{0}  1 
one 

10^{1}  10 
ten 
10^{2}  100 
one hundred 
10^{3}  1,000 
one thousand 
10^{4}  10,000 
ten thousand 
10^{5}  100,000 
one hundred thousand 
10^{6}  1,000,000 
one million 
10^{7}  10,000,000 
ten million 
10^{8}  100,000,000 
one hundred million 
10^{9}  1,000,000,000 
one billion 
10^{10}  10,000,000,000 
ten billion 
10^{11}  100,000,000,000

one hundred billion 
10^{12}  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 tenthousandth 
10^{5}  0.00001 
one hundredthousandth 
10^{6}  0.000001 
one millionth 
10^{7}  0.0000001 
one tenmillionth 
10^{8}  0.00000001 
one hundredmillionth 
10^{9}  0.000000001 
one billionth 
10^{10}  0.0000000001 
one tenbillionth 
10^{11} 
0.00000000001 
one hundredbillionth 
10^{12} 
0.000000000001 
one trillionth 
Table P.2.2: Negative Powers of Ten
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