Given that 𝑥 squared over 𝑦 squared equals 25 over nine, evaluate 𝑥 cubed over 𝑦 cubed.
We need to take 𝑥 squared over 𝑦 squared equals 25 over nine and use that to find out what 𝑥 cubed over 𝑦 cubed would be. To do that, we could first find out what 𝑥 is equal to and what 𝑦 is equal to. Notice that the numerator and the denominator are whole numbers and they’re also square numbers. 25 is equal to five squared and nine is equal to three squared. At this point, it looks like we found 𝑥 and 𝑦. However, the square root of 25 is not just five. Negative five squared also equals positive 25. Negative five times negative five equals positive 25. And the same is true for nine. Negative three squared equals nine. And so, we have to say that 𝑥 equals plus or minus five. And 𝑦 equals plus or minus three.
And so, 𝑥 cubed over 𝑦 cubed could be positive five cubed over positive three cubed. It could be equal to negative five cubed over positive three cubed or positive five cubed over negative three cubed. And the final option would be negative five cubed over negative three cubed. Five cubed equals 125 and three cubed equals 27. 𝑥 cubed over 𝑦 cubed will be equal to plus or minus 125 over 27. We could write that as 125 over 27 and negative 125 over 27.
I wanna show one more way we could’ve solved this problem. If you didn’t recognize these square numbers, you could’ve set 𝑥 squared equal to 25 and 𝑦 squared equal to nine. And then take in the square root of both sides of the equation. The square root of 𝑥 squared equals 𝑥, and the square root of 25 is plus or minus five. The same thing is true for the 𝑦. Take the square root of both sides. The square root of 𝑦 squared equals 𝑦, and the square root of nine equals plus or minus three. And then, you could take that information and plug it back into 𝑥 cubed over 𝑦 cubed. Which would give 125 over 27 or negative 125 over 27.