If `(a + (1)/(a)) ^(2) = 3,` then the value of `a ^(30) + a ^(24) + a ^(18) + a ^(12) + a ^(6) + 1 ` is:

Options

A. `-1`
B. 1
C. 27
D. 0

Answer

To solve the equation \((a + \frac{1}{a})^2 = 3\) and find the value of \(a^{30} + a^{24} + a^{18} + a^{12} + a^{6} + 1\), we can follow these steps:

### Step 1: Simplify the given equation
Starting with the equation:
\[ (a + \frac{1}{a})^2 = 3 \]
Taking the square root of both sides, we get:
\[ a + \frac{1}{a} = \sqrt{3} \quad \text{or} \quad a + \frac{1}{a} = -\sqrt{3} \]

### Step 2: Cube both sides
We will use the positive root for simplicity:
\[ a + \frac{1}{a} = \sqrt{3} \]
Now, we cube both sides:
\[ (a + \frac{1}{a})^3 = (\sqrt{3})^3 \]
Using the identity \( (x + y)^3 = x^3 + y^3 + 3xy(x + y) \), we have:
\[ a^3 + \frac{1}{a^3} + 3(a)(\frac{1}{a})(a + \frac{1}{a}) = 3\sqrt{3} \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} + 3(a + \frac{1}{a}) = 3\sqrt{3} \]
Substituting \(a + \frac{1}{a} = \sqrt{3}\):
\[ a^3 + \frac{1}{a^3} + 3\sqrt{3} = 3\sqrt{3} \]
Thus, we find:
\[ a^3 + \frac{1}{a^3} = 0 \]

### Step 3: Relate \(a^6\) to the equation
From \(a^3 + \frac{1}{a^3} = 0\), we can deduce:
\[ a^3 = -\frac{1}{a^3} \]
Multiplying both sides by \(a^3\):
\[ a^6 + 1 = 0 \implies a^6 = -1 \]

### Step 4: Substitute \(a^6\) into the expression
Now we need to evaluate:
\[ a^{30} + a^{24} + a^{18} + a^{12} + a^{6} + 1 \]
We can express each term in terms of \(a^6\):
\[ a^{30} = (a^6)^5 = (-1)^5 = -1 \]
\[ a^{24} = (a^6)^4 = (-1)^4 = 1 \]
\[ a^{18} = (a^6)^3 = (-1)^3 = -1 \]
\[ a^{12} = (a^6)^2 = (-1)^2 = 1 \]
\[ a^{6} = -1 \]
Now substituting these values into the expression:
\[ -1 + 1 - 1 + 1 - 1 + 1 \]

### Step 5: Simplify the expression
Now we can simplify:
\[ (-1 + 1) + (-1 + 1) + (-1 + 1) = 0 + 0 + 0 = 0 \]

### Final Answer
Thus, the value of \(a^{30} + a^{24} + a^{18} + a^{12} + a^{6} + 1\) is:
\[ \boxed{0} \]