If Z is a positive integer such that \((Z/81)^{1/3} = Y^{4} —7\)

What is the value of Y?

(Statement1): \(Z^{5/2} = 3^{25}\)
—> \(Z = 3^{25*(2/5)}= 3^{10}\)

\((3^{10}/ 81)^{1/3}= Y^{4} —7\)
\(3^{2} = Y^{4} —7\)
\(Y^{4}= 16\)

Y= 2, Y= —2
Insufficient

( Statement2)
—4 < Y—1< 4
—3< Y< 5
Clearly insufficient

Taken together 1&2,
Y=—2 and Y= 2
—3 < Y < 5
Insuffient

The answer is E

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But the question says if z is a positive integer!!
So statement 1 will be sufficient?

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The question says that Z is a positive number, not Y.

The first thing that stands out is that I don't really see a way to simplify the info from the question stem. I could add 7 to both sides, but then I'd need to take a fourth root of both sides to get Y by itself, and that seems to make things more complicated, not less complicated. I could also cube both sides to get rid of the cube root, but that would make the right side really complicated. So, I'll just leave it for now.

The second thing that stands out is that z is a positive integer. However, it never says that y is a positive integer! It also doesn't say that y is an integer at all. I bet that's a trap: the problem writer wants me to assume that y is a positive integer, but there may be negative or non-integer results for y that also work.

Finally, the second statement looks way easier than the first one. So, I'll start there, with the intention of bailing out afterwards if this problem starts to take too long.

Statement 2: |y - 1| < 4. Normally, I would want to call this one insufficient immediately, and keep moving. However, I remember the constraint that says that z has to be a positive integer, and I'm a little bit suspicious. Is it possible that there's only one value of z that even gives us a value of y in this range? If we have time, let's do a little more work with this one.

|y - 1| < 4 means that y is within 4 units of 1 on the number line. In other words, -3 < y < 5.

Are there multiple valid values of y in this range? Or is there perhaps only one? Let's try a few things.

If z = 81, then the equation would simplify like this:

\(\sqrt[3]{\frac{81}{81}} =Y^4−7\)

\(1 =Y^4−7\)

\(Y^4 = 8\)

I don't know exactly what value of Y solves this. However, I'm confident of two things. First, it's definitely between -3 and 5. (It's somewhere between 1 and 2, actually.) Second, there are two different values of Y in that range that solve this equation, because Y can be either positive or negative.

Therefore, Y can have two different values, so the statement is insufficient. Eliminate B and D.

This is more work than we technically needed to do, by the way. On test day, it's probably better to just conclude that it's insufficient without actually proving it with math!

Statement 1: This lets us calculate the exact value of z. However, it doesn't let us calculate the exact value of y. That's because, just like with the other statement, y will have both a positive value and a negative value, due to the 4th power. So, this one is insufficient as well. Eliminate A.

Statements 1 and 2 together:

It's possible that they're sufficient together, but it seems unlikely. On test day, I'd probably just pick E and keep moving.

One way to check would be to figure out the actual two values of y that you get from Statement 1. Then, if both of those values are within the range given in statement 2, then you know the two statements are insufficient put together. That's because you'd have two values of y that work with both statements. Otherwise, we'd need to do a ton more math to find different values, so I'm going to try that before I do anything else...

\(Z^{\frac{5}{2}}=3^{25}\)

\(Z^{5}=3^{50}\)

\(Z = 3^{10}\)

\(\sqrt[3]{\frac{3^{10}}{81}} =Y^4−7\)

\(\sqrt[3]{\frac{3^{10}}{3^{4}}} =Y^4−7\)

\(\sqrt[3]{3^6} =Y^4−7\)

\(3^2 = Y^4 - 7\)

\(Y^4 = 16\)

Y = 2, Y = -2

Okay, there are two values of Y that fit both statements: 2, and -2. Therefore, the statements are insufficient together and the answer is E! (Interestingly, if the question instead said "What is the value of Y^2?", we'd have more work to do...
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