To simplify the complex fraction
\[ \frac{\frac{6}{a^2 b} - \frac{5}{a b^2}}{\frac{3}{a^2 b^2} + \frac{8}{a b}}, \]
we will start by simplifying both the numerator and the denominator separately.
Step 1: Simplifying the Numerator
The numerator is
\[ \frac{6}{a^2 b} - \frac{5}{a b^2}. \]
To combine these fractions, we need a common denominator. The least common denominator (LCD) for \(a^2 b\) and \(a b^2\) is \(a^2 b^2\).
Now, we rewrite each fraction with the common denominator:
\[ \frac{6}{a^2 b} = \frac{6 b}{a^2 b^2} \]
\[ \frac{5}{a b^2} = \frac{5 a}{a^2 b^2} \]
So, we have:
\[ \frac{6 b - 5 a}{a^2 b^2}. \]
Step 2: Simplifying the Denominator
The denominator is
\[ \frac{3}{a^2 b^2} + \frac{8}{a b}. \]
Again, we need a common denominator. The LCD for \(a^2 b^2\) and \(a b\) is \(a^2 b^2\).
Rewriting the fractions:
\[ \frac{3}{a^2 b^2} = \frac{3}{a^2 b^2} \]
\[ \frac{8}{a b} = \frac{8 a b}{a^2 b^2} \]
Now, add these two fractions:
\[ \frac{3 + 8 a b}{a^2 b^2}. \]
Step 3: Putting It All Together
Now we can substitute back into our complex fraction:
\[ \frac{\frac{6b - 5a}{a^2 b^2}}{\frac{3 + 8ab}{a^2 b^2}}. \]
Since the denominators in both the numerator and the denominator are the same (\(a^2 b^2\)), we can simplify:
\[ = \frac{6b - 5a}{3 + 8ab}. \]
Conclusion
Thus, the simplified form of the complex fraction is:
\[ \frac{6b - 5a}{3 + 8ab}. \]