Negative Means Reciprocal
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 28 of 64.
Push the quotient rule one step past zero. x^2 / x^5 cancels two pairs and leaves three factors on the bottom: 1 / x^3. The rule's arithmetic says 2 - 5 = -3, so the same expression is x^-3. Equal again, so: x^-3 = 1/x^3.
A negative exponent is not a negative number in disguise. It is an address: the factors live on the other side of the fraction bar. 2^-3 is 1/2^3 = 1/8, a small positive number.
Descending powers of 2 tell the story in one line: 8, 4, 2, 1, 1/2, 1/4, 1/8. Each step down divides by 2, and the sequence sails through 2^0 = 1 without a bump.
Challenge
EasyRewrite 3^-2 as a fraction, then work it out: type each line.
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