Speak Math Now!/Week 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
- a^{m} • a^{n} = a^{m + 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 = x^{5}
- b^{2} • b^{5} = b^{7}
Rule 2 (Power to a Power)Edit
- (a^{m})^{n} = a^{m • n}
Exponents with an exponent: multiply exponents.
-if you have multiple numbers in your parenthesis, all numbers get the exponent.
- Examples
- (d^{5})^{3} = d^{15}
- (xyz)^{2} = x^{2}y^{2}z^{2}
- (xyz^{2})^{3} = x^{3}y^{3}z^{6}
Rule 3 (Multiple Power Rules)Edit
As the title says, multiple power rules
Rule #1 + #2 in the same problem:
- Examples
- 6(x^{3}y)(x^{2}y)^{3}
- 6(x^{3}y)(x^{6}y^{3})
- 6x^{9}y^{4}
Rule 4 (Quotient of Powers)Edit
Divide numbers, subtract exponents
- Examples
- a^{m}/a^{n}= a^{m-n}
- x^{4}/x^{2}=x^{2}
- 6x^{7}y^{3}/12x^{3}y^{8}
- [divide 6 and 12 by "6", minus the exponents ^{7} and ^{3}, minus the exponents ^{3} and ^{8}].
- 1x^{4}/2y^{5}
- [clean up the 1--DON'T INCLUDE 1!]
- x^{4}/2y^{5}
Rule 5 (Power of a Quotient)Edit
An exponent affects all...
- Examples
- (a/b)^{m} = a^{m}/b^{m}
- (2a^{3}b^{5}/3b^{2})^{2}
- (2^{2}a^{6}b^{10}/3^{2}b^{4})
- [All numbers get affected, including exponents!]
- (4a^{6}b^{10}/9b^{4})
- [There are two "b"s, so you have to subtract b^{10} by b^{4}... because the result is positive (b^{6}), this number goes in the numerator space]
- (4a^{6}b^{6}/9)
Rule 6 (Negative Exponents)Edit
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/a^{n}
- 1/a^{-n} = a^{n}/1
- 4a^{-3}b^{6}/16x^{2}b^{-2}
- 4b^{2}b^{6}/16x^{2}a^{3}
- 4b^{8}/16x^{2}a^{3}
- b^{8}/4x^{2}a^{3}
The base is best left positiveEdit
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, B^{1/2}= is an imaginary number. It can also be shown that is also imaginary.^{[1]}
QuizEdit
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 |
LogarithmsEdit
Visit the optional subpage /Logarithms to learn about the related logarithm function. |