If `x^(2)+((1)/(x^(2)))=1`, then what is the value of `x^(48)+x^(42)+x^(36)+x^(30)+x^(24)+x^(18)+x^(12)+x^(8)+1?`

Options

A. `-9`
B. `0`
C. `1`
D. `9`

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 1 \) and find the value of \( x^{48} + x^{42} + x^{36} + x^{30} + x^{24} + x^{18} + x^{12} + x^{8} + 1 \), we can proceed as follows:

### Step 1: Simplify the given equation
We start with the equation:
\[ x^2 + \frac{1}{x^2} = 1 \]
We can rewrite this as:
\[ x^2 + \frac{1}{x^2} - 1 = 0 \]
Let \( y = x + \frac{1}{x} \). Then we know that:
\[ x^2 + \frac{1}{x^2} = y^2 - 2 \]
Thus, we can substitute:
\[ y^2 - 2 = 1 \implies y^2 = 3 \implies y = \sqrt{3} \text{ or } y = -\sqrt{3} \]

### Step 2: Find \( x + \frac{1}{x} \)
From our previous step, we have:
\[ x + \frac{1}{x} = \sqrt{3} \text{ or } x + \frac{1}{x} = -\sqrt{3} \]

### Step 3: Find \( x^3 + \frac{1}{x^3} \)
Using the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x}) \]
Substituting \( y = \sqrt{3} \):
\[ x^3 + \frac{1}{x^3} = (\sqrt{3})^3 - 3\sqrt{3} = 3\sqrt{3} - 3\sqrt{3} = 0 \]

### Step 4: Find \( x^6 + \frac{1}{x^6} \)
Using the identity:
\[ x^6 + \frac{1}{x^6} = (x^3 + \frac{1}{x^3})^2 - 2 \]
Substituting \( x^3 + \frac{1}{x^3} = 0 \):
\[ x^6 + \frac{1}{x^6} = 0^2 - 2 = -2 \]

### Step 5: Find \( x^{12} + \frac{1}{x^{12}} \)
Using the same identity:
\[ x^{12} + \frac{1}{x^{12}} = (x^6 + \frac{1}{x^6})^2 - 2 \]
Substituting \( x^6 + \frac{1}{x^6} = -2 \):
\[ x^{12} + \frac{1}{x^{12}} = (-2)^2 - 2 = 4 - 2 = 2 \]

### Step 6: Find \( x^{24} + \frac{1}{x^{24}} \)
Using the same identity:
\[ x^{24} + \frac{1}{x^{24}} = (x^{12} + \frac{1}{x^{12}})^2 - 2 \]
Substituting \( x^{12} + \frac{1}{x^{12}} = 2 \):
\[ x^{24} + \frac{1}{x^{24}} = 2^2 - 2 = 4 - 2 = 2 \]

### Step 7: Find \( x^{48} + \frac{1}{x^{48}} \)
Using the same identity:
\[ x^{48} + \frac{1}{x^{48}} = (x^{24} + \frac{1}{x^{24}})^2 - 2 \]
Substituting \( x^{24} + \frac{1}{x^{24}} = 2 \):
\[ x^{48} + \frac{1}{x^{48}} = 2^2 - 2 = 4 - 2 = 2 \]

### Step 8: Calculate the final expression
Now we can calculate:
\[ x^{48} + x^{42} + x^{36} + x^{30} + x^{24} + x^{18} + x^{12} + x^{8} + 1 \]
We know:
- \( x^{48} + \frac{1}{x^{48}} = 2 \)
- \( x^{42} + \frac{1}{x^{42}} = 0 \) (since \( x^{42} = x^{36} \cdot x^6 \) and \( x^{36} + \frac{1}{x^{36}} = 2 \))
- \( x^{36} + \frac{1}{x^{36}} = 2 \)
- \( x^{30} + \frac{1}{x^{30}} = 0 \)
- \( x^{24} + \frac{1}{x^{24}} = 2 \)
- \( x^{18} + \frac{1}{x^{18}} = 0 \)
- \( x^{12} + \frac{1}{x^{12}} = 2 \)
- \( x^{8} + \frac{1}{x^{8}} = 0 \)
- \( 1 = 1 \)

Adding these values:
\[ 2 + 0 + 2 + 0 + 2 + 0 + 2 + 0 + 1 = 9 \]

### Final Answer
Thus, the value of \( x^{48} + x^{42} + x^{36} + x^{30} + x^{24} + x^{18} + x^{12} + x^{8} + 1 \) is \( \boxed{9} \).