Algebra 1/Unit 9: Six rules of Exponents
Subject classification: this is a mathematics resource. |
Educational level: this is a secondary education resource. |
Type classification: this is a lesson resource. |
Completion status: this resource is ~75% complete. |
Rule 1 (Product of Powers)
[edit | edit source]- am • an = am + n
Multiply exponents with the same base - add exponents
- Examples
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.
- Examples
- (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:
- Examples
- 6(x3y)(x2y)3
- 6(x3y)(x6y3)
- 6x9y4
Rule 4 (Quotient of Powers)
[edit | edit source]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
Rule 5 (Power of a Quotient)
[edit | edit source]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)
[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
- 4a-3b6/16x2b-2
- 4b2b6/16x2a3
- 4b8/16x2a3
- b8/4x2a3
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:
Here:
- 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.[1]
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. |