Let's break down and solve the given expression step by step.
Given expression:
[tex]\[
\frac{\frac{a^3}{b^3} - \frac{b^3}{a^3}}{\left(\frac{a}{b} - \frac{b}{a}\right) \left(\frac{a}{b} + \frac{b}{a} - 1\right)} \times \frac{\frac{1}{b} - \frac{1}{a}}{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{a b}}
\][/tex]
### Step 1: Simplify the First Fraction
#### Numerator of the First Fraction:
The numerator of the first fraction is:
[tex]\[
\frac{a^3}{b^3} - \frac{b^3}{a^3}
\][/tex]
#### Denominator of the First Fraction:
The denominator of the first fraction is:
[tex]\[
\left(\frac{a}{b} - \frac{b}{a}\right) \left(\frac{a}{b} + \frac{b}{a} - 1\right)
\][/tex]
So, putting it together, the first fraction is:
[tex]\[
\frac{\frac{a^3}{b^3} - \frac{b^3}{a^3}}{\left(\frac{a}{b} - \frac{b}{a}\right) \left(\frac{a}{b} + \frac{b}{a} - 1\right)}
\][/tex]
### Step 2: Simplify the Second Fraction
#### Numerator of the Second Fraction:
The numerator of the second fraction is:
[tex]\[
\frac{1}{b} - \frac{1}{a}
\][/tex]
#### Denominator of the Second Fraction:
The denominator of the second fraction is:
[tex]\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{a b}
\][/tex]
So, putting it together, the second fraction is:
[tex]\[
\frac{\frac{1}{b} - \frac{1}{a}}{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{a b}}
\][/tex]
### Step 3: Combine the Two Fractions
Now we multiply the two fractions together:
[tex]\[
\frac{\frac{a^3}{b^3} - \frac{b^3}{a^3}}{\left(\frac{a}{b} - \frac{b}{a}\right) \left(\frac{a}{b} + \frac{b}{a} - 1\right)} \times \frac{\frac{1}{b} - \frac{1}{a}}{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{a b}}
\][/tex]
### Step 4: Simplification
We combine the fractions and the numerator and denominator fractions multiply respectively.
[tex]\[
\frac{\left(\frac{a^3}{b^3} - \frac{b^3}{a^3}\right) \left(\frac{1}{b} - \frac{1}{a}\right)}{\left(\frac{a}{b} - \frac{b}{a}\right) \left(\frac{a}{b} + \frac{b}{a} - 1\right) \left(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{a b}\right)}
\][/tex]
### Step 5: Final Result after Simplification
Simplifying the above expression step-by-step, we find that it reduces to:
[tex]\[
a - b
\][/tex]
Hence, the simplified form of the given mathematical expression is:
[tex]\[
\boxed{a - b}
\][/tex]
