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Simplify the following expression:

[tex]\[
\frac{\frac{a^3}{b^3}-\frac{b^3}{a^3}}{\left(\frac{a}{b}-\frac{b}{a}\right)\left(\frac{a}{b}+\frac{b}{a}-1\right)} \times \frac{\frac{1}{b}-\frac{1}{a}}{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{ab}}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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