# Algebra 1/Unit 9: Six rules of Exponents

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Educational level: this is a secondary education resource. |

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## Rule 1 (Product of Powers)

[edit | edit source]- 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 | edit source]- (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 | edit source]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 | edit source]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}].

- [divide 6 and 12 by "6", minus the exponents
- 1x
^{4}/2y^{5}- [clean up the 1--DON'T INCLUDE 1!]

- x
^{4}/2y^{5}

## Rule 5 (Power of a Quotient)

[edit | edit source]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]

- [There are two "b"s, so you have to subtract b
- (4a
^{6}b^{6}/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/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 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, B
^{1/2}= 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 |