If -1 < X < Y < 0, which of the following expressions are listed in order from least to greatest?
Explanation
Given -1 < X < Y < 0, both X and Y are negative.
Let's analyze the expressions:
xy²: Since y² is positive (because squaring a number makes it positive), and x is negative, xy² is negative.
x²y: Since x² is positive, and y is negative, x²y is negative.
xy: Since both x and y are negative, xy is positive.
Comparing the signs:
xy is positive.
Both xy² and x²y are negative.
Since xy is positive and the others are negative, xy must be the greatest.
Now, comparing xy² and x²y:
Let's consider the magnitude:
|x| > |y| (since X < Y < 0)
xy² = x * y² (x is more negative, y² is positive)
x²y = x² * y (x² is positive, y is negative)
Given |x| > |y|, x² > y² (since both are positive after squaring)
x²|y| > |x|y² (since |y| < |x| and both x² and y² are positive)
x²y < xy² (both are negative, but x²y is more negative)
So, the order from least to greatest is:
x²y, xy², xy
