Answer :
We begin with the expression
[tex]$$
\frac{x^7 - x^3}{4x^9 - 4x^5}.
$$[/tex]
Step 1. Factor the numerator.
Notice that both terms in the numerator have a common factor of [tex]$x^3$[/tex]. Factor this out:
[tex]$$
x^7 - x^3 = x^3(x^4 - 1).
$$[/tex]
Step 2. Factor the denominator.
Similarly, factor out the common factor [tex]$4x^5$[/tex] in the denominator:
[tex]$$
4x^9 - 4x^5 = 4x^5(x^4 - 1).
$$[/tex]
Step 3. Write the expression in factored form.
Substitute the factored forms of the numerator and denominator into the original expression:
[tex]$$
\frac{x^3(x^4 - 1)}{4x^5(x^4 - 1)}.
$$[/tex]
Step 4. Cancel common factors.
The factor [tex]$(x^4 - 1)$[/tex] appears in both the numerator and the denominator, so we cancel it (provided that [tex]$x^4 - 1 \neq 0$[/tex]):
[tex]$$
\frac{x^3 \,\cancel{(x^4 - 1)}}{4x^5 \,\cancel{(x^4 - 1)}} = \frac{x^3}{4x^5}.
$$[/tex]
Step 5. Simplify the remaining expression.
Reduce [tex]$\frac{x^3}{4x^5}$[/tex] by dividing both the numerator and the denominator by [tex]$x^3$[/tex], noting that [tex]$x \neq 0$[/tex]:
[tex]$$
\frac{x^3}{4x^5} = \frac{1}{4x^2}.
$$[/tex]
Final Answer:
[tex]$$
\frac{1}{4x^2}.
$$[/tex]
Thus, the simplified form of the expression
[tex]$$
\frac{x^7 - x^3}{4x^9 - 4x^5}
$$[/tex]
is
[tex]$$
\frac{1}{4x^2}.
$$[/tex]
[tex]$$
\frac{x^7 - x^3}{4x^9 - 4x^5}.
$$[/tex]
Step 1. Factor the numerator.
Notice that both terms in the numerator have a common factor of [tex]$x^3$[/tex]. Factor this out:
[tex]$$
x^7 - x^3 = x^3(x^4 - 1).
$$[/tex]
Step 2. Factor the denominator.
Similarly, factor out the common factor [tex]$4x^5$[/tex] in the denominator:
[tex]$$
4x^9 - 4x^5 = 4x^5(x^4 - 1).
$$[/tex]
Step 3. Write the expression in factored form.
Substitute the factored forms of the numerator and denominator into the original expression:
[tex]$$
\frac{x^3(x^4 - 1)}{4x^5(x^4 - 1)}.
$$[/tex]
Step 4. Cancel common factors.
The factor [tex]$(x^4 - 1)$[/tex] appears in both the numerator and the denominator, so we cancel it (provided that [tex]$x^4 - 1 \neq 0$[/tex]):
[tex]$$
\frac{x^3 \,\cancel{(x^4 - 1)}}{4x^5 \,\cancel{(x^4 - 1)}} = \frac{x^3}{4x^5}.
$$[/tex]
Step 5. Simplify the remaining expression.
Reduce [tex]$\frac{x^3}{4x^5}$[/tex] by dividing both the numerator and the denominator by [tex]$x^3$[/tex], noting that [tex]$x \neq 0$[/tex]:
[tex]$$
\frac{x^3}{4x^5} = \frac{1}{4x^2}.
$$[/tex]
Final Answer:
[tex]$$
\frac{1}{4x^2}.
$$[/tex]
Thus, the simplified form of the expression
[tex]$$
\frac{x^7 - x^3}{4x^9 - 4x^5}
$$[/tex]
is
[tex]$$
\frac{1}{4x^2}.
$$[/tex]
