Why Exponents Subtract
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 12 of 64.
The exponents subtract because the factors cancel in pairs. To see this, write x^6 / x^2 out in full: six factors of x on top and two factors of x on the bottom. The fraction x/x equals 1 whenever x is not zero. So each factor on the bottom pairs with one factor on top, and the pair becomes a factor of 1. Multiplying by 1 does not change the product.
Two pairs cancel, and four factors of x are left on top, so x^6 / x^2 = x^4. The subtraction 6 - 2 does not subtract the values of the powers. It counts the factors that are left after the pairing.
The same pairing explains two cases you will meet later. One is when the top and the bottom have the same number of factors. The other is when the bottom has more factors than the top. Both come in the chapter on zero and negative exponents.
Challenge
EasySimplify x^8 / x^5 / 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