Ratio and Proportion:
Ratio and proportion are important mathematical concepts used to compare and relate quantities and values. In this problem, we are given a ratio of x to y, which is 3:4. We need to find the value of a given expression.

Using the Ratio:
To solve the problem, we need to use the given ratio to find the values of x and y. We can do this by assuming a common multiplier k, which will give us:

x = 3k
y = 4k

Substituting the Values:
Now, we can substitute these values into the given expression and simplify it. We get:

x^2 y = (3k)^2 * 4k = 36k^3
xy^2 = 3k * (4k)^2 = 48k^3
x^3 y^3 = (3k)^3 * (4k)^3 = 1728k^6

Final Answer:
Therefore, the value of the expression x^2 y : xy^2 : x^3 y^3 is:

36k^3 : 48k^3 : 1728k^6
Simplifying this, we get:

3:4:144

Conclusion:
In conclusion, we can say that the value of the expression x^2 y : xy^2 : x^3 y^3 is 3:4:144. We used the given ratio to find the values of x and y and then substituted them into the expression to simplify it.