Let's break this problem into steps for clarity. Here's the given mathematical expression:

$\left( \left( \frac{6}{7} \right)^{-2} - \left( \frac{4}{5} \right)^0 \right) : \left( \left( \frac{1}{13} \right)^{-1} \right)$

Step 1: Simplify each part

(a) $\left( \frac{6}{7} \right)^{-2}$

Using the negative exponent rule, $a^{-n} = \frac{1}{a^n}$: $\left( \frac{6}{7} \right)^{-2} = \frac{1}{\left( \frac{6}{7} \right)^2} = \frac{1}{\frac{36}{49}} = \frac{49}{36}.$

(b) $\left( \frac{4}{5} \right)^0$

Any number raised to the power of 0 equals 1: $\left( \frac{4}{5} \right)^0 = 1.$

(c) $\left( \frac{1}{13} \right)^{-1}$

Using the negative exponent rule, $a^{-1} = \frac{1}{a}$: $\left( \frac{1}{13} \right)^{-1} = \frac{1}{\frac{1}{13}} = 13.$

Step 2: Substitute simplified values into the expression

Now substitute the simplified values: $\left( \frac{49}{36} - 1 \right) : 13.$

Step 3: Simplify the numerator

$\frac{49}{36} - 1 = \frac{49}{36} - \frac{36}{36} = \frac{13}{36}.$

Step 4: Divide by 13

The expression becomes: $\frac{\frac{13}{36}}{13} = \frac{13}{36} \times \frac{1}{13} = \frac{1}{36}.$

Final Answer:

$\boxed{\frac{1}{36}}$

Let me know if you want further details or have additional questions!

Relative Questions:

1. What are the properties of exponents, and how are they applied here?
2. How do you handle operations with fractions during simplifications?
3. Why does any number raised to the power of zero equal 1?
4. How does dividing by a number relate to multiplying by its reciprocal?
5. Can we extend these rules to negative or fractional exponents?

Tip: Always handle negative exponents and operations step-by-step to avoid errors.