The problem asks us to simplify the expression $\sqrt[5]{-1024x^5y^6}$, assuming all variables are nonnegative real numbers.

1. Problem Description

The problem asks us to simplify the expression $\sqrt[5]{-1024x^5y^6}$, assuming all variables are nonnegative real numbers.

2. Solution Steps

First, we can rewrite the expression as:

$\sqrt[5]{-1024x^5y^6} = \sqrt[5]{-1 \cdot 1024 \cdot x^5 \cdot y^5 \cdot y}$

We know that $1024 = 2^{10} = (2^2)^5 = 4^5$, so

$\sqrt[5]{-1024x^5y^6} = \sqrt[5]{-1 \cdot 4^5 \cdot x^5 \cdot y^5 \cdot y}$

Now we can take the fifth root:

$\sqrt[5]{-1 \cdot 4^5 \cdot x^5 \cdot y^5 \cdot y} = \sqrt[5]{-1} \cdot \sqrt[5]{4^5} \cdot \sqrt[5]{x^5} \cdot \sqrt[5]{y^5} \cdot \sqrt[5]{y}$

Since $\sqrt[5]{-1} = -1$

$\sqrt[5]{4^5} = 4$

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

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

we can simplify this to:

$-1 \cdot 4 \cdot x \cdot y \cdot \sqrt[5]{y} = -4xy\sqrt[5]{y}$

3. Final Answer

The simplified expression is $-4xy\sqrt[5]{y}$.