Answer :
To simplify the expression [tex]\(\frac{x^7 - x^3}{4x^9 - 4x^5}\)[/tex], we can follow these steps:
1. Factor the Numerator:
The numerator is [tex]\(x^7 - x^3\)[/tex]. Notice that both terms have a common factor of [tex]\(x^3\)[/tex]. So, we can factor it out:
[tex]\[
x^7 - x^3 = x^3(x^4 - 1)
\][/tex]
2. Factor the Denominator:
The denominator is [tex]\(4x^9 - 4x^5\)[/tex]. Here, both terms have a common factor of [tex]\(4x^5\)[/tex]. Factor it out as follows:
[tex]\[
4x^9 - 4x^5 = 4x^5(x^4 - 1)
\][/tex]
3. Simplify the Expression:
After factoring, the fraction becomes:
[tex]\[
\frac{x^3(x^4 - 1)}{4x^5(x^4 - 1)}
\][/tex]
Now, we can cancel out the common factor [tex]\((x^4 - 1)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{x^3}{4x^5}
\][/tex]
4. Simplify Further:
Next, simplify the fraction by cancelling out powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{4x^5} = \frac{x^{3-5}}{4} = \frac{1}{4x^2}
\][/tex]
So, the simplified expression is [tex]\(\frac{1}{4x^2}\)[/tex].
1. Factor the Numerator:
The numerator is [tex]\(x^7 - x^3\)[/tex]. Notice that both terms have a common factor of [tex]\(x^3\)[/tex]. So, we can factor it out:
[tex]\[
x^7 - x^3 = x^3(x^4 - 1)
\][/tex]
2. Factor the Denominator:
The denominator is [tex]\(4x^9 - 4x^5\)[/tex]. Here, both terms have a common factor of [tex]\(4x^5\)[/tex]. Factor it out as follows:
[tex]\[
4x^9 - 4x^5 = 4x^5(x^4 - 1)
\][/tex]
3. Simplify the Expression:
After factoring, the fraction becomes:
[tex]\[
\frac{x^3(x^4 - 1)}{4x^5(x^4 - 1)}
\][/tex]
Now, we can cancel out the common factor [tex]\((x^4 - 1)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{x^3}{4x^5}
\][/tex]
4. Simplify Further:
Next, simplify the fraction by cancelling out powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{4x^5} = \frac{x^{3-5}}{4} = \frac{1}{4x^2}
\][/tex]
So, the simplified expression is [tex]\(\frac{1}{4x^2}\)[/tex].
