The Zero Exponent
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 26 of 64.
The quotient rule walks straight into a strange door. x^7 / x^7 is any nonzero number divided by itself, so its value is certainly 1. But the rule says: subtract the exponents, 7 - 7 = 0, giving x^0.
Both answers describe the same expression, so they must be equal: x^0 = 1 for any nonzero x. The zero exponent is not an arbitrary convention; it is the only value that keeps the quotient rule honest.
Zero factors of x multiply to the empty product, and the empty product is 1 for the same reason an empty sum is 0: it is the starting point that changes nothing.
Challenge
EasySimplify x^3 * x^2 / x^5.
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