Sure, let's work through the simplification of the given expression step by step:
Given expression:
[tex]\[ \frac{n^5 - a^5}{n^4 - a^4} \][/tex]
We need to handle this in a way that allows us to simplify the fraction. Let's begin by factoring both the numerator and the denominator.
1. Factor the numerator [tex]\(n^5 - a^5\)[/tex]:
- This is a difference of powers, which can be factored using the formula for the difference of powers:
[tex]\[ n^5 - a^5 = (n - a)(n^4 + n^3a + n^2a^2 + na^3 + a^4) \][/tex]
2. Factor the denominator [tex]\(n^4 - a^4\)[/tex]:
- This is a difference of squares, which we can rewrite and factor further:
[tex]\[ n^4 - a^4 = (n^2)^2 - (a^2)^2 = (n^2 - a^2)(n^2 + a^2) \][/tex]
- Now, factor the difference of squares [tex]\(n^2 - a^2\)[/tex]:
[tex]\[ n^2 - a^2 = (n - a)(n + a) \][/tex]
- Combine these factors:
[tex]\[ n^4 - a^4 = (n - a)(n + a)(n^2 + a^2) \][/tex]
So, substituting the factored forms back into the original expression, we get:
[tex]\[ \frac{(n - a)(n^4 + n^3a + n^2a^2 + na^3 + a^4)}{(n - a)(n + a)(n^2 + a^2)} \][/tex]
3. Simplify the expression by canceling common factors:
- The term [tex]\((n - a)\)[/tex] appears in both the numerator and the denominator. As long as [tex]\(n \neq a\)[/tex], we can cancel these terms:
[tex]\[ \frac{n^4 + n^3a + n^2a^2 + na^3 + a^4}{(n + a)(n^2 + a^2)} \][/tex]
So, the simplified form of the expression [tex]\(\frac{n^5 - a^5}{n^4 - a^4}\)[/tex] is:
[tex]\[ \frac{a^5 - n^5}{a^4 - n^4} \][/tex]
