College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Simplify the expression:

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

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].