Explain the meaning of the exponent 3 in the algebraic expression (x+y) ^3

Since the exponent is 3, and in the parentheses is x+y, it's basically

(x+y) (x+y) (x+y)

You write out x+y 3 times since the exponent is 3.

If the exponent was 4, you would write out x+y 4 times and so on.

The 3 gives value to the x+y and how many to use or calculate before getting your answer.