To find the value of [tex]\( x^3 + \frac{1}{x^3} \)[/tex] where [tex]\( x = 3 + \sqrt{8} \)[/tex], we can proceed step-by-step as follows:
1. Calculate [tex]\( x \)[/tex]:
[tex]\[
x = 3 + \sqrt{8}
\][/tex]
Given in the result:
[tex]\[
x \approx 5.82842712474619
\][/tex]
2. Calculate [tex]\( x^3 \)[/tex]:
[tex]\[
x^3 = (3 + \sqrt{8})^3
\][/tex]
The result provides:
[tex]\[
x^3 \approx 197.99494936611663
\][/tex]
3. Calculate [tex]\( \frac{1}{x} \)[/tex]:
[tex]\[
\frac{1}{x} = \frac{1}{3 + \sqrt{8}}
\][/tex]
The result provides:
[tex]\[
\frac{1}{x} \approx 0.1715728752538099
\][/tex]
4. Calculate [tex]\( \left(\frac{1}{x}\right)^3 \)[/tex]:
[tex]\[
\left(\frac{1}{x}\right)^3 = \frac{1}{x^3}
\][/tex]
The result provides:
[tex]\[
\left(\frac{1}{x}\right)^3 \approx 0.005050633883346584
\][/tex]
5. Sum [tex]\( x^3 \)[/tex] and [tex]\( \left(\frac{1}{x}\right)^3 \)[/tex]:
[tex]\[
x^3 + \frac{1}{x^3} = 197.99494936611663 + 0.005050633883346584
\][/tex]
Therefore:
[tex]\[
x^3 + \frac{1}{x^3} \approx 197.99999999999997
\][/tex]
So, the value of [tex]\( x^3 + \frac{1}{x^3} \)[/tex] is approximately [tex]\( 197.99999999999997 \)[/tex], which is very close to 198.
