Algebra 1/Unit 9: Six rules of Exponents
Rule 1 (Product of Powers)[edit | edit source]
- 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)[edit | edit source]
- (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)[edit | edit source]
As the title says, multiple power rules
Rule #1 + #2 in the same problem:
Rule 4 (Quotient of Powers)[edit | edit source]
Divide numbers, subtract exponents
- am/an= am-n
- [divide 6 and 12 by "6", minus the exponents 7 and 3, minus the exponents 3 and 8].
- [clean up the 1--DON'T INCLUDE 1!]
Rule 5 (Power of a Quotient)[edit | edit source]
An exponent affects all...
- (a/b)m = am/bm
- [All numbers get affected, including exponents!]
- [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]
Rule 6 (Negative Exponents)[edit | edit source]
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
The base is best left positive[edit | edit source]
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:
- 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.
- 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.
Quiz[edit | edit source]
If you would like to take the quiz on the Six rules of Exponents, please go to Speak Math Now!/Week 9: Six rules of Exponents/Quiz
Logarithms[edit | edit source]
Visit the optional subpage /Logarithms to learn about the related logarithm function.