Product Rule or Power Rule?
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 18 of 64.
Two rules, one confusion. x^4 * x^3 multiplies two different groups of factors standing side by side: the counts add, x^7. (x^4)^3 makes three copies of one group: the counts multiply, x^12.
The test is the shape. Side by side with a multiplication sign between the powers: add. Nested, one power inside a parenthesis with an exponent outside: multiply. Say what the expression is doing before touching the exponents.
When in doubt, unpack a small case. x^2 * x^2 is four factors; (x^2)^2 is also four. The two rules agree exactly where the shapes coincide, and nowhere else.
Challenge
EasySimplify x^6 * (x^2)^3. Which rule comes first?
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