Let's solve the given expression step by step.
The expression we need to simplify is:
[tex]\[
\frac{\frac{a^3}{b^3}-\frac{b^3}{a^3}}{\left(\frac{a}{b}-\frac{b}{a}\right)\left(\frac{a}{b}+\frac{b}{a}-1\right)} \times \frac{\frac{1}{b}-\frac{1}{a}}{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{a b}}
\][/tex]
### Step 1: Simplify the first fraction
#### Numerator of the first fraction:
[tex]\[
\frac{a^3}{b^3} - \frac{b^3}{a^3}
\][/tex]
#### Denominator of the first fraction:
[tex]\[
\left( \frac{a}{b} - \frac{b}{a} \right) \left( \frac{a}{b} + \frac{b}{a} - 1 \right)
\][/tex]
### Step 2: Simplify the second fraction
#### Numerator of the second fraction:
[tex]\[
\frac{1}{b} - \frac{1}{a}
\][/tex]
#### Denominator of the second fraction:
[tex]\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{ab}
\][/tex]
Now, we handle each part separately.
### Simplify the first fraction:
#### Numerator:
[tex]\[
\frac{a^3}{b^3} - \frac{b^3}{a^3} = \frac{a^6 - b^6}{a^3 b^3}
\][/tex]
Observe:
[tex]\[
a^6 - b^6 = (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3)
\][/tex]
So, the numerator becomes:
[tex]\[
\frac{(a^3 - b^3)(a^3 + b^3)}{a^3 b^3}
\][/tex]
#### Denominator:
The first part:
[tex]\[
\frac{a}{b} - \frac{b}{a} = \frac{a^2 - b^2}{ab}
\][/tex]
The second part:
[tex]\[
\frac{a}{b} + \frac{b}{a} - 1 = \frac{a^2 + b^2}{ab} - 1 = \frac{a^2 + b^2 - ab}{ab}
\][/tex]
So, the denominator becomes:
[tex]\[
\left(\frac{a^2 - b^2}{ab}\right) \left(\frac{a^2 + b^2 - ab}{ab}\right) = \frac{(a^2 - b^2)(a^2 + b^2 - ab)}{a^2 b^2}
\][/tex]
Putting it all together, the first fraction is:
[tex]\[
\frac{\frac{(a^3 - b^3)(a^3 + b^3)}{a^3 b^3}}{\frac{(a^2 - b^2)(a^2 + b^2 - ab)}{a^2 b^2}} = \frac{(a^3 - b^3)(a^3 + b^3)}{(a^2 - b^2)(a^2 + b^2 - ab)}
\][/tex]
### Simplify the second fraction:
#### Numerator:
[tex]\[
\frac{1}{b} - \frac{1}{a} = \frac{a - b}{ab}
\][/tex]
#### Denominator:
[tex]\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{ab} = \frac{b^2 + a^2 + ab}{a^2 b^2}
\][/tex]
So, the second fraction becomes:
[tex]\[
\frac{\frac{a - b}{ab}}{\frac{a^2 + b^2 + ab}{a^2 b^2}} = \frac{a - b}{ab} \times \frac{a^2 b^2}{a^2 + b^2 + ab} = \frac{(a - b) a b}{a^2 + b^2 + ab}
\][/tex]
### Combine both fractions:
Combining the results of the simplified first and second fractions we get:
[tex]\[
\left(\frac{(a^3 - b^3)(a^3 + b^3)}{(a^2 - b^2)(a^2 + b^2 - ab)}\right) \times \left(\frac{(a - b) a b}{a^2 + b^2 + ab}\right)
\][/tex]
This simplifies to:
[tex]\[
\frac{(a^3 - b^3)(a^3 + b^3) (a - b) a b}{(a^2 - b^2)(a^2 + b^2 - ab)(a^2 + b^2 + ab)}
\][/tex]
Cancel out common factors, we reach the simplified form:
[tex]\[
a - b
\][/tex]
So, the final simplified expression is:
[tex]\[
\boxed{a - b}
\][/tex]
