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

1. Problem Description

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

2. Solution Steps

First, rewrite 1024 as $2^{10}$.

$\sqrt[5]{1024x^{10}y^{11}} = \sqrt[5]{2^{10}x^{10}y^{11}}$

We can rewrite this as:

$\sqrt[5]{2^{10}x^{10}y^{10}y}$

Separate the terms:

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

$= \sqrt[5]{(2^2x^2y^2)^5y}$

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

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

The expression $\sqrt[n]{a^n}$ simplifies to $a$ if $n$ is odd and $a$ is any real number or if $n$ is even and $a$ is non-negative. Here the index is 5 which is odd and the variables are non-negative real numbers.

3. Final Answer

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