Algebra 1/Unit 9: Six rules of Exponents

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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.

  • 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.

  • (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:

  • 6(x3y)(x2y)3
  • 6(x3y)(x6y3)
  • 6x9y4

Rule 4 (Quotient of Powers)


Divide numbers, subtract exponents

  • 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

Rule 5 (Power of a Quotient)


An exponent affects all...

  • (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 is the base of the exponential expression, with the requirements that B≠1 and 0<B<.
  • 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: −<X<.
  • 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=−1, B1/2=  is an imaginary number. It can also be shown that   is also imaginary.[1]





  Visit the optional subpage /Logarithms to learn about the related logarithm function.


  1.   (Just Google it!)