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David Terr
Ph.D. Math, UC Berkeley

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.

100 101 1 one 10 ten 100 one hundred 1,000 one thousand 10,000 ten thousand 100,000 one hundred thousand 1,000,000 one million 10,000,000 ten million 100,000,000 one hundred million 1,000,000,000 one billion 10,000,000,000 ten billion 100,000,000,000 one hundred billion 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 10-2 0.1 one tenth 0.01 one hundredth 0.001 one thousandth 0.0001 one ten-thousandth 1e-05 one hundred-thousandth 1e-06 one millionth 1e-07 one ten-millionth 1e-08 one hundred-millionth 1e-09 one billionth 1e-10 one ten-billionth 1e-11 one hundred-billionth 1e-12 one trillionth

Table P.2.2: Negative Powers of Ten