Counts below zero
Walk down the powers of 5: 25, 5, 1. What is the next step?
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5² = 25. Step 1 of 5
5²= 25.5¹= 5. One step down divides by 5.5⁰= 1. Divide by 5 again.5⁻¹= 1/5. Divide by 5 again.5⁻²= 1/25. Divide by 5 again.
The rule
The next step is 1 ÷ 5 = 1/5, which is 5⁻¹.
The ladder does not stop at zero.
Rule: Each step down divides by the base. A count of −2 divides 1 by the base twice.
In symbols, n⁻ᵃ = 1/nᵃ, for any base n that is not 0. Here a is a positive whole number.
The top count may be smaller than the bottom count. 5² ÷ 5⁴ = 5⁻².
Counts still add. 5⁶ × 5⁻⁴ = 5², because 6 + (−4) = 2. Check: 15,625 ÷ 625 = 25.
3⁻³= 1 ÷ 3 ÷ 3 ÷ 3 = 1/2710⁻²= 1 ÷ 10 ÷ 10 = 1/100
Watch out!
The minus in the count adds no minus to the number. 5⁻² is 1/25, not −25.