Why Exponents Multiply
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 17 of 64.
Slow the rule down once, for good. (x^2)^4 is four copies of x^2: x^2 * x^2 * x^2 * x^2. Now the product rule from chapter two takes over: same base, add the exponents, 2 + 2 + 2 + 2 = 8. Adding the same number four times is multiplication: 2 * 4.
So the power of a power rule is not a new fact about exponents; it is the product rule applied to identical packages, with the repeated addition collapsed into a product. Each law in this section leans on the previous one.
(x^2)^4 = x^8, and the same argument gives the general sentence you will write shortly: packages of m factors, n packages, mn factors in all.
Challenge
EasyWrite (x^3)^3 as a single power of x.
Try it yourself
Simplify the expression
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