To find the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] given that [tex]\( x + \frac{1}{x} = 9 \)[/tex], we will proceed step-by-step through algebraic manipulations.
1. Step 1: Find [tex]\( x^2 + \frac{1}{x^2} \)[/tex]
Start by squaring both sides of the given equation:
[tex]\[
\left( x + \frac{1}{x} \right)^2 = 9^2
\][/tex]
Expanding the left side using the binomial theorem:
[tex]\[
x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81
\][/tex]
Simplify the expression:
[tex]\[
x^2 + 2 + \frac{1}{x^2} = 81
\][/tex]
Subtract 2 from both sides:
[tex]\[
x^2 + \frac{1}{x^2} = 79
\][/tex]
2. Step 2: Find [tex]\( x^4 + \frac{1}{x^4} \)[/tex]
Next, square the expression [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[
\left( x^2 + \frac{1}{x^2} \right)^2 = 79^2
\][/tex]
Expand the left side:
[tex]\[
x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} = 6241
\][/tex]
Simplify the expression:
[tex]\[
x^4 + 2 + \frac{1}{x^4} = 6241
\][/tex]
Subtract 2 from both sides:
[tex]\[
x^4 + \frac{1}{x^4} = 6239
\][/tex]
Thus, the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] is
[tex]\[
\boxed{6239}
\][/tex]
