Algebra 1/Unit 9: Six rules of Exponents

Rule 1 (Product of Powers)

 * am • an = am + n

Multiply exponents with the same base - add exponents Here, we will list examples of this rule. If you have any questions on how some of these examples have been done, please go to the talk page.
 * Examples
 * x • xxxx = x5
 * b2 • b5 = b7

Rule 2 (Power to a Power)

 * (am)n = am • n

Exponents with an exponent: multiply exponents. -if you have multiple numbers in your parenthesis, all numbers get the exponent.
 * Examples:
 * (d5)3 = d15
 * (xyz)2 = x2y2z2
 * (xyz2)3 = x3y3z6

Rule 3 (Multiple Power Rules)
As the title says, multiple power rules Rule #1 + #2 in the same problem:
 * Examples:
 * 6(x3y)(x2y)3
 * 6(x3y)(x6y3)
 * 6x9y4

Rule 4 (Quotient of Powers)
Divide numbers, subtract exponents
 * Examples
 * am/an= am-n
 * x4/x2=x2
 * 6x7y3/12x3y8
 * [divide 6 and 12 by "6", minus the exponents 7 and 3, minus the exponents 3 and 8].
 * 1x4/2y5
 * [clean up the 1--DON'T INCLUDE 1!]
 * x4/2y5
 * x4/2y5

Rule 5 (Power of a Quotient)
An exponent affects all...


 * Examples
 * (a/b)m = am/bm
 * (2a3b5/3b2)2
 * (22a6b10/32b4)
 * [All numbers get affected, including exponents!]
 * (4a6b10/9b4)
 * [There are two "b"s, so you have to subtract b10 by b4... because the result is positive (b6), this number goes in the numerator space]
 * (4a6b6/9)

Rule 6 (Negative Exponents)
This does not apply to negative numbers, but exponents! As discussed in the next section, it is common to demand that the base of an exponential function is a positive number (that does not equal 1.)


 * a-n = 1/an
 * 1/a-n = an/1

-


 * 4a-3b6/16x2b-2
 * 4b2b6/16x2a3
 * 4b8/16x2a3
 * b8/4x2a3

The base is best left positive
Review. To identify the three parts to an exponential expression, consider the case where B=2, X=3, and V=8:

$$B^X= 2^3 = 8 =V$$

Here:
 * B is the base of the exponential expression, with the requirements that B≠1 and 0&lt;B&lt; &infin;.
 * X is the expression's exponent. While the consequences of letting X have an imaginary part are fascinating, this discussion considers only the case where X is a real number: &minus; &infin; &lt;X&lt; &infin;.
 * V is the value of the exponential expression.

Two comments are in order:
 * 1) Though our choice of "X" and "V" as variables is not unusual, there is nothing "standard" about this notation. On the other hand, it is common to use the lower-case "b" to denote the base.  The capital "B" was used here it because the lower-case "b" was used a number of times in the preceding examples.
 * 2) It is worth mentioning why we exclude negative values of B from any discussion where we wish to all X to range over all positive and negative values on the real axis. It is well-known that for B=&minus;1, B1/2=$$\sqrt{-1}=i$$ is an imaginary number. It can also be shown that $$\sqrt{-2}$$ is also imaginary.