Sure, let's work through the problem step-by-step.
### Given:
[tex]\[ x = 3 + \sqrt{D} \][/tex]
For the purpose of this problem, let's assume [tex]\( D = 0 \)[/tex].
### Step 1: Calculate [tex]\( x \)[/tex]
Since [tex]\( D = 0 \)[/tex]:
[tex]\[ x = 3 + \sqrt{0} \][/tex]
[tex]\[ x = 3 \][/tex]
### Step 2: Calculate [tex]\( x^2 \)[/tex]
We know [tex]\( x = 3 \)[/tex], so:
[tex]\[ x^2 = 3^2 \][/tex]
[tex]\[ x^2 = 9 \][/tex]
### Step 3: Calculate [tex]\( \frac{1}{x^3} \)[/tex]
Next, we need to find [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 = 3^3 \][/tex]
[tex]\[ x^3 = 27 \][/tex]
Now, calculate [tex]\( \frac{1}{x^3} \)[/tex]:
[tex]\[ \frac{1}{x^3} = \frac{1}{27} \][/tex]
[tex]\[ \frac{1}{x^3} \approx 0.037037 \][/tex]
### Step 4: Calculate the final result
We are asked to find:
[tex]\[ x^2 + \frac{1}{x^3} \][/tex]
Substitute the values we found:
[tex]\[ x^2 + \frac{1}{x^3} = 9 + 0.037037 \][/tex]
Combine these values:
[tex]\[ x^2 + \frac{1}{x^3} \approx 9.037037 \][/tex]
### Summary of Results:
- [tex]\( x \approx 3.0 \)[/tex]
- [tex]\( x^2 \approx 9.0 \)[/tex]
- [tex]\( \frac{1}{x^3} \approx 0.037037 \)[/tex]
- [tex]\( x^2 + \frac{1}{x^3} \approx 9.037037 \)[/tex]
Thus, the value of [tex]\( x^2 + \frac{1}{x^3} \)[/tex] is approximately [tex]\( 9.037037 \)[/tex].
