To find the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] given that [tex]\( x + \frac{1}{x} = 9 \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
x + \frac{1}{x} = 9
\][/tex]
2. Square both sides of the equation to eliminate the fraction:
[tex]\[
\left( x + \frac{1}{x} \right)^2 = 9^2
\][/tex]
3. Simplify the left-hand side:
[tex]\[
x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81
\][/tex]
Since [tex]\( x \cdot \frac{1}{x} = 1 \)[/tex], we have:
[tex]\[
x^2 + 2 + \frac{1}{x^2} = 81
\][/tex]
4. Subtract 2 from both sides to isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[
x^2 + \frac{1}{x^2} = 79
\][/tex]
5. Square the new equation to find [tex]\( x^4 + \frac{1}{x^4} \)[/tex]:
[tex]\[
\left( x^2 + \frac{1}{x^2} \right)^2 = 79^2
\][/tex]
6. Simplify the left-hand side:
[tex]\[
x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} = 6241
\][/tex]
Again, [tex]\( x^2 \cdot \frac{1}{x^2} = 1 \)[/tex], so we have:
[tex]\[
x^4 + 2 + \frac{1}{x^4} = 6241
\][/tex]
7. Subtract 2 from both sides to isolate [tex]\( x^4 + \frac{1}{x^4} \)[/tex]:
[tex]\[
x^4 + \frac{1}{x^4} = 6239
\][/tex]
Hence, the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] is [tex]\( 6239 \)[/tex].
