Answer :

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.