Apply the properties of integer exponents to simplify the algebraic expression and it is in parentheses 2XY small three on top of the Y and then we have outside of the parentheses a small for parentheses and with a small on top and why and then outside of the parentheses we had a three on top With the answer be on top small 15 on top or would it be 16 X with a small 25 on top or 32 X with a small tan on top and white with a small 11 on top or 16 X with a small tan on top with a small 15 on top

Answer 1

To simplify the expression you provided, we need to carefully interpret the notation. From your description, it seems like we have the expression:

\[ (2XY^3)^4 \]

This expression is raised to the power of 4, and we can apply the properties of exponents to simplify it.

Using the property \((ab)^n = a^n b^n\) and \((x^m)^n = x^{m \cdot n}\), we can break down the expression as follows:

1. Raise the coefficient and each variable inside the parentheses to the power of 4:

   \[ (2XY^3)^4 = 2^4 \cdot X^4 \cdot (Y^3)^4 \]

2. Calculate \(2^4\):

   \[ 2^4 = 16 \]

3. For \(X^4\), it remains \(X^4\).

4. For \(Y^3\) raised to the power of 4, we use the property \((Y^3)^4 = Y^{3 \cdot 4} = Y^{12}\).

Putting it all together, we have:

\[ (2XY^3)^4 = 16X^4Y^{12} \]

If we wanted to express this with some additional notation you mentioned, but based on the final result, the expression simplifies to:

\[ 16X^4Y^{12} \]

As for the suggested options you mentioned like \(16X^{25}\), \(32X^{10}Y^{11}\), or \(16X^{10}\) with the numerator seemingly being some form, they don't align with the simplified expression we've derived from the exponentiation.

Thus, the correct answer is:

\[ \boxed{16X^4Y^{12}} \]