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INTEGER EXPONENTS AND SCIENTIFIC NOTATIONIn this section, we will introduce another symbol for a fraction of the form and then we will use this symbol to write certain numbers in simpler form. INTEGER EXPONENTS Recall that we have defined a power a^{n} (where n is a natural number) as follows: a^{n} = a · a · a · · · · · a (n factors) We will now give meaning to powers in which the exponent is 0 or a negative integer. First, let us consider the quotient a^{4}/a^{4}. Using the property of quotients of powers, we have Note that for any a not equal to zero, the lefthand member equals 1 and the righthand member equals a^{0}. In general, we define: a^{0} = 1 for any number a except 0. Example 1 a. 3^{0} = 1 Now consider the quotient a^{4}/a^{7}. Using the two quotient laws for powers, we have Thus, for any a not equal to 0, we can view a^{3} as equivalent to . In general, we define for any number a except zero. Example 2 SCIENTIFIC NOTATION Very large numbers such as 5,980,000,000,000, 000,000,000,000,000 and very small numbers such as 0.000 000 000 000 000 000 000 001 67 occur in many scientific areas. Large numbers can be rewritten in a more compact and useful form by using powers with positive exponents. We can also rewrite small numbers by using powers with negative exponents that have been introduced in this section. First, let us consider some factored forms of 38,400 in which one of the factors is a power of 10. Although any one of such factored forms may be more useful than the original form of the number, a special name is given to the last form. A number expressed as the product of a number between 1 and 10 (including 1) and a power of 10 is said to be in scientific form or scientific notation. For example, 4.18 x 10^{4} , 9.6 x 10^{2}, and 4 x 10^{5} are in scientific form. Now, let us consider some factored forms of 0.0057 in which one of the factors is a power of 10. In this case, 5.7 x 10^{3} is the scientific form for 0.0057. To write a number in scientific form:
Example 3 If a number is written in scientific form and we want to rewrite it in standard form, we simply reverse the above procedure. Example 4 Common Error: Note that The exponent only applies to the x, not the 3. Thus, CHAPTER SUMMARY
