We need to simplify the expression $\sqrt[5]{1024x^{10}y^{11}}$, assuming that all variables are nonnegative real numbers.

1. Problem Description

We need to simplify the expression $\sqrt[5]{1024x^{10}y^{11}}$, assuming that all variables are nonnegative real numbers.

2. Solution Steps

We want to simplify the expression $\sqrt[5]{1024x^{10}y^{11}}$.

First, we express 1024 as a power of 2: $1024 = 2^{10}$.

So, we have $\sqrt[5]{2^{10}x^{10}y^{11}}$.

Next, we rewrite the expression using fractional exponents:

$(2^{10}x^{10}y^{11})^{\frac{1}{5}}$

Then we apply the power rule:

$2^{\frac{10}{5}}x^{\frac{10}{5}}y^{\frac{11}{5}}$

Simplify the exponents:

$2^2 x^2 y^{\frac{11}{5}}$

Rewrite $y^{\frac{11}{5}}$ as $y^{\frac{10}{5} + \frac{1}{5}} = y^{\frac{10}{5}} y^{\frac{1}{5}} = y^2 y^{\frac{1}{5}}$.

The expression now is:

$2^2 x^2 y^2 y^{\frac{1}{5}}$

$2^2 = 4$, so we have

$4x^2y^2 y^{\frac{1}{5}}$

Rewrite $y^{\frac{1}{5}}$ as $\sqrt[5]{y}$.

So, the final expression is:

$4x^2y^2 \sqrt[5]{y}$

3. Final Answer

$4x^2y^2\sqrt[5]{y}$