Why Exponents Multiply
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 17 of 64.
The exponents multiply because the product rule is applied to identical copies. (x^2)^4 is four copies of x^2 multiplied together: x^2 * x^2 * x^2 * x^2. All four copies have the same base, so the product rule adds the exponents: 2 + 2 + 2 + 2 = 8. Adding the same number four times is the same as multiplying it by 4, which is 2 * 4.
So the power of a power rule comes from the product rule. You apply the product rule to identical copies of a power, and the repeated addition becomes a multiplication.
The result is (x^2)^4 = x^8. The same reasoning works for any exponents. If each copy holds m factors and there are n copies, there are 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