To simplify the given expression
[tex]\[
\frac{\frac{4}{x-8} + 1}{x + \frac{16}{x-8}}
\][/tex]
follow these detailed steps:
1. Define a substitution: Let [tex]\( y = x - 8 \)[/tex]. This simplifies the expression involving [tex]\( x \)[/tex].
2. Rewrite the numerator:
[tex]\[
\frac{4}{x-8} + 1 = \frac{4}{y} + 1
\][/tex]
Convert 1 to a fraction with the same denominator [tex]\( y \)[/tex]:
[tex]\[
\frac{4}{y} + 1 = \frac{4}{y} + \frac{y}{y} = \frac{4 + y}{y}
\][/tex]
3. Rewrite the denominator:
[tex]\[
x + \frac{16}{x-8} = y + 8 + \frac{16}{y}
\][/tex]
Combine the terms into a single fraction:
[tex]\[
y + \frac{8y + 16}{y} = \frac{y^2}{y} + \frac{8y}{y} + \frac{16}{y} = \frac{y^2 + 8y + 16}{y}
\][/tex]
4. Simplify the quadratic expression in the denominator:
[tex]\[
y^2 + 8y + 16 = (y + 4)^2
\][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[
\frac{\frac{4 + y}{y}}{\frac{y^2 + 8y + 16}{y}} = \frac{4 + y}{(y + 4)^2}
\][/tex]
6. Apply the original substitution back (remember [tex]\( y = x - 8 \)[/tex]):
[tex]\[
\frac{4 + (x - 8)}{((x - 8) + 4)^2} = \frac{x - 4}{(x - 4)^2}
\][/tex]
7. Simplify the fraction:
[tex]\[
\frac{x - 4}{(x - 4)^2} = \frac{1}{x - 4}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{1}{x - 4}}
\][/tex]
