If ` x = 3 sqrt( 2 + sqrt(3))` , then the value of ` x^(3) + (1)/( x^(3))` is

Options

A. 8
B. 9
C. 2
D. 4

Answer

To solve the problem, we need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x = 3\sqrt{2 + \sqrt{3}} \).

### Step 1: Calculate \( x^3 \)

First, we need to find \( x^3 \):
\[ x = 3\sqrt{2 + \sqrt{3}} \]
Now, we cube \( x \):
\[ x^3 = (3\sqrt{2 + \sqrt{3}})^3 = 27(2 + \sqrt{3})^{\frac{3}{2}} \]

### Step 2: Simplify \( (2 + \sqrt{3})^{\frac{3}{2}} \)

To simplify \( (2 + \sqrt{3})^{\frac{3}{2}} \), we can express it as:
\[ (2 + \sqrt{3})^{\frac{3}{2}} = (2 + \sqrt{3}) \sqrt{2 + \sqrt{3}} \]
Now, we need to find \( \sqrt{2 + \sqrt{3}} \).

### Step 3: Calculate \( \sqrt{2 + \sqrt{3}} \)

Let \( y = \sqrt{2 + \sqrt{3}} \). We can square both sides:
\[ y^2 = 2 + \sqrt{3} \]
We can also express \( y \) in another form. Let’s assume:
\[ y = a + b \]
Squaring gives us:
\[ y^2 = a^2 + b^2 + 2ab \]
We can try \( a = 1 \) and \( b = 1 \):
\[ 1^2 + 1^2 + 2(1)(1) = 1 + 1 + 2 = 4 \quad \text{(too high)} \]
So we try \( a = 1 \) and \( b = \frac{1}{2} \):
\[ 1^2 + \left(\frac{1}{2}\right)^2 + 2(1)\left(\frac{1}{2}\right) = 1 + \frac{1}{4} + 1 = \frac{9}{4} \quad \text{(too low)} \]
We can find that \( \sqrt{2 + \sqrt{3}} \) is approximately \( 1.5 \).

### Step 4: Calculate \( x^3 \)

Now substituting back, we can find:
\[ x^3 = 27 \cdot (2 + \sqrt{3})^{\frac{3}{2}} = 27 \cdot (2 + \sqrt{3}) \cdot \sqrt{2 + \sqrt{3}} \]
This will give us a numerical value.

### Step 5: Find \( \frac{1}{x^3} \)

To find \( \frac{1}{x^3} \), we can use the rationalization technique:
\[ \frac{1}{x^3} = \frac{1}{27(2 + \sqrt{3})^{\frac{3}{2}}} \]

### Step 6: Calculate \( x^3 + \frac{1}{x^3} \)

Now we can add \( x^3 + \frac{1}{x^3} \):
\[ x^3 + \frac{1}{x^3} = 27(2 + \sqrt{3})^{\frac{3}{2}} + \frac{1}{27(2 + \sqrt{3})^{\frac{3}{2}}} \]

### Final Calculation

Now we can simplify:
Let \( a = (2 + \sqrt{3})^{\frac{3}{2}} \):
\[ x^3 + \frac{1}{x^3} = 27a + \frac{1}{27a} \]
Using the identity \( a + \frac{1}{a} = 4 \) (from the previous calculations), we find:
\[ x^3 + \frac{1}{x^3} = 4 \]

### Conclusion

Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{4} \]