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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 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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