[tex] - 10 \frac{4}{5} \div (1 \div \frac{10}{17})[/tex]

First, let's simplify the expression inside the parentheses:

[tex]1 \div \frac{10}{17} = 1 \times \frac{17}{10} = \frac{17}{10}[/tex]

Now, we have:

[tex]-10 \frac{4}{5} \div \frac{17}{10}[/tex]

Next, let's convert the mixed number to an improper fraction:

[tex]-10 \frac{4}{5} = -\frac{54}{5}[/tex]

Now, we have:

[tex]-\frac{54}{5} \div \frac{17}{10}[/tex]

To divide by a fraction, we multiply by the reciprocal:

[tex]-\frac{54}{5} \times \frac{10}{17}= -\frac{540}{85} = -\frac{108}{17}[/tex]

Therefore,

[tex]-10 \frac{4}{5} \div (1 \div \frac{10}{17}) = -\frac{108}{17}[/tex]