The Rule in Letters
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 9 of 64.
Every product of same-base powers you have combined follows one rule: a^m × a^n = a^(m+n). The letters make the rule general. a is any base, and m and n are any exponents. The equation says that the counts of factors add, whatever they are.
The rule in letters can be checked against its reason. a^m is m factors of a, and a^n is n more. Putting them in a row gives m + n factors. The letters do not add new mathematics. They state the same counting argument once for all cases.
Learn to recognize rules in this general form. The rest of this section states each law this way. From here on, you will also use the laws on expressions where the exponents themselves are letters.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Exponents & Polynomials
1What a Power Is
Repeated MultiplicationSquares and CubesPowers of Negative NumbersWhere the Minus LivesRecap: First Powers4Power of a Power
A Power Raised AgainWhy Exponents MultiplyProduct Rule or Power Rule?The Rule in LettersRecap: Power of a Power2The Product Rule
Multiplying Same BasesWith CoefficientsMore Than Two FactorsThe Rule in LettersRecap: The Product Rule5Product and Quotient Powers
Every Factor Gets the PowerWith a Number InsideA Power of a QuotientAll Three Rules TogetherRecap: Products and Quotients8What a Polynomial Is
Terms with PowersThe DegreeLike Terms, Only StrongerRecap: Meet the Polynomials11Dividing by a Monomial
Splitting the FractionDividing by a Variable TermBigger SplitsCancel with CareRecap: Dividing3The Quotient Rule
Dividing Same BasesWhy Exponents SubtractWith CoefficientsThe Rule in LettersRecap: The Quotient Rule6Zero and Negative Exponents
The Zero ExponentZero at WorkNegative Means ReciprocalWriting Without NegativesPowers of TenRecap: Zero and Negative