Answer :

Sure, let's simplify the given expression step-by-step:

Given Expression:

[tex]\[
\frac{x^7 - x^3}{4x^9 - 4x^5}
\][/tex]

Step 1: Factoring the Numerator and Denominator

First, we'll factor common terms from the numerator and the denominator.

- Numerator: [tex]\(x^7 - x^3\)[/tex]

We can factor out [tex]\(x^3\)[/tex]:

[tex]\[
x^7 - x^3 = x^3(x^4 - 1)
\][/tex]

- Denominator: [tex]\(4x^9 - 4x^5\)[/tex]

We can factor out [tex]\(4x^5\)[/tex]:

[tex]\[
4x^9 - 4x^5 = 4x^5(x^4 - 1)
\][/tex]

Step 2: Simplifying the Expression

Now, our expression with factored terms looks like this:

[tex]\[
\frac{x^3(x^4 - 1)}{4x^5(x^4 - 1)}
\][/tex]

Notice that [tex]\((x^4 - 1)\)[/tex] is a common factor in both the numerator and the denominator. We can cancel this common factor out:

[tex]\[
\frac{x^3}{4x^5}
\][/tex]

Step 3: Simplifying Further

Now, we simplify [tex]\(\frac{x^3}{4x^5}\)[/tex]:

- [tex]\(x^3\)[/tex] over [tex]\(x^5\)[/tex] simplifies by subtracting the exponents:

[tex]\[
\frac{x^3}{x^5} = \frac{1}{x^{5-3}} = \frac{1}{x^2}
\][/tex]

So the expression further simplifies to:

[tex]\[
\frac{1}{4x^2}
\][/tex]

Final Simplified Expression:

The simplified form of the given expression is:

[tex]\[
\frac{1}{4x^2}
\][/tex]

This is the simplest form of the original expression.

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