Fractions and Negatives
Part of the Exponentials, Logs & Sequences section of Coddy's Algebra 2 journey. Lesson 14 of 49.
A negative exponent means a reciprocal: 3^(-1) = 1/3. So a fraction like 1/3 is a power of 3, and powers of 1/3 are powers of 3 with the sign of the exponent flipped. For example, (1/3)^(-2) is 3^2, which is 9.
Use that to solve (1/3)^x = 9. Write 9 as a power of 1/3: 9 = (1/3)^(-2). The equation reads (1/3)^x = (1/3)^(-2), the same base on both sides, so x = -2. Check: (1/3)^(-2) flips to 3^2, which is 9.
The negative answer makes sense. Positive powers of 1/3 shrink below 1, so to land on 9, above 1, the exponent has to be negative.
Challenge
Solve (1/2)^x = 16. Write 16 as a power of 1/2, then set the exponents equal.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Exponentials, Logs & Sequences
1Exponential Functions
Repeated MultiplyingGrowth or DecayDrawing GrowthDrawing DecayRecap: Exponential Functions3Same-Base Equations
Matching PowersRewriting the BaseAn Expression UpstairsFractions and NegativesRecap: Same Base