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Study Guides > College Algebra CoRequisite Course

The Rules for Exponents

Learning Outcomes

  • Recall the properties of exponents and use them to rewrite expressions containing exponents.
Review the following list of rules for simplifying expressions containing exponents, then try the problems listed below. If you need a refresher, return to the Algebra Essentials module for more explanation and demonstration.

The Product Rule of Exponents

For any real number aa and natural numbers mm and nn, the product rule of exponents states that
aman=am+n{a}^{m}\cdot {a}^{n}={a}^{m+n}

The Quotient Rule of Exponents

For any real number aa and natural numbers mm and nn, such that m>nm>n, the quotient rule of exponents states that
aman=amn\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}

The Power Rule of Exponents

For any real number aa and positive integers mm and nn, the power rule of exponents states that
(am)n=amn{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}

The Zero Exponent Rule of Exponents

For any nonzero real number aa, the zero exponent rule of exponents states that
a0=1{a}^{0}=1

The Negative Rule of Exponents

For any nonzero real number aa and natural number nn, the negative rule of exponents states that
an=1an and an=1an{a}^{-n}=\dfrac{1}{{a}^{n}} \text{ and } {a}^{n}=\dfrac{1}{{a}^{-n}}

The Power of a Product Rule of Exponents

For any real numbers aa and bb and any integer nn, the power of a product rule of exponents states that
(ab)n=anbn\large{\left(ab\right)}^{n}={a}^{n}{b}^{n}

The Power of a Quotient Rule of Exponents

For any real numbers aa and bb and any integer nn, the power of a quotient rule of exponents states that
(ab)n=anbn\large{\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}

Try it

[ohm_question]52398[/ohm_question] [ohm_question]52400[/ohm_question] [ohm_question]14054[/ohm_question] [ohm_question]23844[/ohm_question]

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