Let's simplify the expression step by step. The given expression is:

$\frac{3x^{10} y^{-10} z^{-5}}{4 \sqrt{3} \cdot 40x^{12} y^7 z^3}$

To simplify this expression, follow these steps:

Step 1: Simplify the Denominator

Rewrite the denominator by breaking it into parts:

$4 \sqrt{3} \cdot 40x^{12} y^7 z^3 = (4 \cdot 40) \sqrt{3} \cdot x^{12} y^7 z^3 = 160 \sqrt{3} \cdot x^{12} y^7 z^3$

Step 2: Combine Terms in the Fraction

Now, the expression becomes:

$\frac{3x^{10} y^{-10} z^{-5}}{160 \sqrt{3} \cdot x^{12} y^7 z^3}$

Step 3: Simplify the Coefficients

The coefficient in the numerator is $3$ and in the denominator is $160 \sqrt{3}$. Dividing 3 by 160 gives:

$\frac{3}{160 \sqrt{3}}$

To rationalize the denominator, multiply the top and bottom by $\sqrt{3}$:

$\frac{3 \sqrt{3}}{160 \cdot 3} = \frac{\sqrt{3}}{160}$

Step 4: Simplify the Variables Using Exponent Rules

Apply exponent subtraction to each variable:

1. For $x$: $x^{10 - 12} = x^{-2} = \frac{1}{x^2}$
2. For $y$: $y^{-10 - 7} = y^{-17} = \frac{1}{y^{17}}$
3. For $z$: $z^{-5 - 3} = z^{-8} = \frac{1}{z^8}$

Step 5: Combine All Simplified Parts

Now we can rewrite the entire expression as:

$\frac{\sqrt{3}}{160 x^2 y^{17} z^8}$

Final Answer

$\frac{\sqrt{3}}{160 x^2 y^{17} z^8}$

This is the simplified form of the original expression.

Would you like further details or have questions on any specific step?

Here are five related questions to explore:

1. How do you rationalize a denominator involving square roots?
2. What are the exponent rules for dividing terms with the same base?
3. How would the simplification differ if there was a cube root instead of a square root?
4. How do you handle negative exponents in fractions?
5. What are the implications of simplifying expressions with mixed radicals and variables?

Tip: When simplifying complex fractions, focus on one part (numerator, denominator, coefficients, or variables) at a time to avoid mistakes.