If y^k/y^5k = m, what is k?

Great question, Bunuel!!

Target question: What is k?

Given: \(\frac{y^k}{y^{5k}} = y^m\)
IMPORTANT: many people will mistakenly apply some exponent laws and conclude that -4k = m. HOWEVER, before we apply these exponent laws, we must be certain that the base does not equal 0, 1 or -1, in which case the exponent laws fly out the window.

Statement 1: m = 4
Let's TEST some values.

There are several values of y, k and m that satisfy statement 1. Here are two:
Case a: y = 1, k = 1 and m = 4. Plugging this into our given equation, we get \(\frac{1^1}{1^5} = 1^4\), which works! In this case, k = 1

Case b: y = 1, k = 2 and m = 4. Plugging this into our given equation, we get \(\frac{1^2}{1^{10}} = 1^4\), which works! In this case, k = 2

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y = 4
There are several values of x and y that satisfy statement 2. Here are two:
Case a: y = 4, k = 1 and m = -4. Plugging this into our given equation, we get \(\frac{4^1}{4^5} = 4^{-4}\), which works! In this case, k = 1

Case b: y = 4, k = 2 and m = -8. Plugging this into our given equation, we get \(\frac{4^2}{4^{10}} = 4^{-8}\), which works! In this case, k = 2

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Since statement 2 tells us that y does NOT equal 0, 1 or -1, we can safely apply some exponent laws:
Given: \(\frac{y^k}{y^{5k}} = y^m\)
Simplify left side to get: \(y^{-4k} = y^m\)
So, -4k = m
Statement 1 tells us that m = 4
Plug this into our equation to get: -4k = 4. which means k must equal -1
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer:
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