1 Exponents
We use the base-10 numbering system, where each place represents a power of 10. Places to the right of the decimal are called decimal places.
Numbers that are very big or very small are easily represented in scientific notation, which multiplies a number between 1 and 9 by 10 with a power or exponent. In the number below, the exponent is 3.
4.321 x 103
Positive exponents represent big numbers (numbers greater than 1) and tell how many times the 10 is multiplied by itself.
100 = 1 (anything to the zero power is 1)
101 = 10 (anything to the one power is itself)
102 = 10 x 10 = 100
103 = 10 x 10 x 10 = 1000
Notice that the number in the positive exponent is equal to the number of zeros in the number being represented.
A negative exponent means to take 1 over the number represented by the positive exponent. This is called the reciprocal.
10-1 = [latex]\frac{1}{10^1}=\frac{1}{10}[/latex] = 0.1
10-2 = [latex]\frac{1}{10^2}=\frac{1}{100}[/latex] = 0.01
10-3 = [latex]\frac{1}{10^3}=\frac{1}{1000}[/latex] = 0.001
Therefore, negative exponents represent small numbers (numbers less than 1). Notice that the number in the negative exponent is one more than the number of zeros between the decimal point and the 1 in the number being represented.
We can use this information to simplify complex fractions as shown in the following example.
Example 1.1
Simplify [latex]\frac{1}{10^{-2}}[/latex]
Answer: Since a negative exponent means inverse
[latex]\frac{1}{10^{-2}}=\frac{1}{1/10^2}[/latex]
Dividing is the same as multiplying by the inverse, so we get
[latex]\frac{1}{1/10^2}=1*\frac{10^2}{1}=10^2[/latex]
Therefore, a negative exponent in the denominator can be rewritten as a positive exponent in the numerator (and vice versa).
