The given expression is:

$\left( 8^{-1} - 9^{-1} \right)^{-1} \div \left( 4^{-1} - 9^{-1} \right)^{-1}$

Step-by-step Solution:

1. Simplify the first term $\left( 8^{-1} - 9^{-1} \right)^{-1}$:

   • $8^{-1} = \frac{1}{8}$ and $9^{-1} = \frac{1}{9}$.
   • So, $8^{-1} - 9^{-1} = \frac{1}{8} - \frac{1}{9}$.
   • Find the common denominator: $\frac{1}{8} - \frac{1}{9} = \frac{9 - 8}{72} = \frac{1}{72}.$
   • Now, $\left( \frac{1}{72} \right)^{-1} = 72$.

2. Simplify the second term $\left( 4^{-1} - 9^{-1} \right)^{-1}$:

   • $4^{-1} = \frac{1}{4}$ and $9^{-1} = \frac{1}{9}$.
   • So, $4^{-1} - 9^{-1} = \frac{1}{4} - \frac{1}{9}$.
   • Find the common denominator: $\frac{1}{4} - \frac{1}{9} = \frac{9 - 4}{36} = \frac{5}{36}.$
   • Now, $\left( \frac{5}{36} \right)^{-1} = \frac{36}{5}$.

3. Divide the two results:

   • The expression becomes: $\frac{72}{\frac{36}{5}} = 72 \times \frac{5}{36} = 10.$

So, the value of the expression is 10.

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