M12-32

Official Solution:


If \(x^3*y^4 = 2000\), what is the value of \(y\)?

Let's begin by factorizing the expression: \(x^3*y^4 = 2^4*5^3\). Although it may seem that \(y=2\) and \(x=5\) would be the solution, it's important to note that we are not given any information about whether \(x\) and \(y\) are integers. Therefore, for any positive value of \(x\), there will exist two corresponding values of \(y\) that satisfy the equation. For instance:

When \(x = \frac{1}{2}\), we get \(y = \sqrt[4]{16000}\) or \(y = -\sqrt[4]{16000}\).

When \(x = 1\), we get \(y = \sqrt[4]{2000}\) or \(y = -\sqrt[4]{2000}\).

When \(x = 2\), we get \(y = \sqrt[4]{250}\) or \(y = -\sqrt[4]{250}\).

...

Similarly, for any nonzero value of \(y\), there will exist corresponding values of \(x\) that satisfy the equation. For instance:

When \(y = \frac{1}{2}\), we get \(x = \sqrt[3]{32000}\).

When \(y = -1\), we get\(x = \sqrt[3]{2000}\).

When \(y = \sqrt{10}\), we get \(x = \sqrt[3]{20}\).

...

Moreover, even if we were given that \(x\) and \(y\) are integers, we cannot deduce that \(y=2\) and \(x=5\) because of the even power of \(y\). It could also be that \(y=-2\).

(1) \(x\) is an integer.

As explained earlier, knowing that \(x\) is an integer does not provide us with sufficient information to determine a unique value for \(y\). Not sufficient.

(2) \(y\) is an integer.

As explained earlier, knowing that \(y\) is an integer does not provide us with sufficient information to determine a unique value for \(y\). Not sufficient.

(1)+(2) We know that both \(x\) and \(y\) are integers. However, as we discussed earlier, this still does not provide us with sufficient information to determine a unique value for \(y\). For instance, it could be that \(y=2\) and \(x=5\), or it could be that \(y=-2\) and \(x=5\). Not sufficient.


Answer: E