Question: 0.999 to the Power of k

Comment on 0.999 to the Power of k

Would the answer be A if it was given that k is a negative integer? (instead of positive as in the given question)
greenlight-admin's picture

Yes, that's correct.

As you'll see in the negative exponents lesson (, there's a nice conversion for negative exponents: (a/b)^(-n) = (b/a)^n
So, for example, (3/5)^(-2) = (5/3)^2 = 25/9

So, in the original question, 0.999 can be rewritten as 999/1000

If the exponent, k, is a negative integer, we can see a pattern emerging:
If k = -1, then (999/1000)^(-1) = (1000/999)^1 = 1000/999
If k = -2, then (999/1000)^(-2) = (1000/999)^2 = 1000²/999²
If k = -3, then (999/1000)^(-3) = (1000/999)^3 = 1000³/999³
We can see that the resulting fraction will ALWAYS have a numerator that is GREATER THAN the denominator.
In other words, the resulting fraction (in Quantity A) will always be greater than 1.

As such, Quantity A will ALWAYS be greater than 0.001 (Quantity B)

Can K also be zero?
greenlight-admin's picture

No, k cannot be zero, because we're told that k is a positive integer.

ASIDE: zero is neither negative nor positive.


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