If ` ( x^(24) + 1)/( x^(12)) = 7` then the value of ` ( x^(72) + 1)/( x^(36)) ` is

Options

A. 433
B. 322
C. 343
D. 432

Answer

To solve the equation \( \frac{x^{24} + 1}{x^{12}} = 7 \) and find the value of \( \frac{x^{72} + 1}{x^{36}} \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ \frac{x^{24} + 1}{x^{12}} = 7 \]
This can be rewritten as:
\[ x^{24} + 1 = 7x^{12} \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ x^{24} - 7x^{12} + 1 = 0 \]

### Step 3: Let \( y = x^{12} \)
Now, we can substitute \( y = x^{12} \):
\[ y^2 - 7y + 1 = 0 \]

### Step 4: Solve the quadratic equation
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ y = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ y = \frac{7 \pm \sqrt{49 - 4}}{2} \]
\[ y = \frac{7 \pm \sqrt{45}}{2} \]
\[ y = \frac{7 \pm 3\sqrt{5}}{2} \]

### Step 5: Find \( x^{36} \)
Since \( y = x^{12} \), we need to find \( x^{36} \):
\[ x^{36} = (x^{12})^3 = y^3 \]
Using \( y = \frac{7 \pm 3\sqrt{5}}{2} \), we can find \( y^3 \).

### Step 6: Calculate \( \frac{x^{72} + 1}{x^{36}} \)
Now we need to find:
\[ \frac{x^{72} + 1}{x^{36}} = \frac{(x^{36})^2 + 1}{x^{36}} = x^{36} + \frac{1}{x^{36}} \]
Let \( z = x^{36} \). Then we need to find \( z + \frac{1}{z} \).

### Step 7: Calculate \( z + \frac{1}{z} \)
Using the identity \( z^2 - 7z + 1 = 0 \) from earlier, we can express \( z + \frac{1}{z} \) in terms of \( z \):
\[ z + \frac{1}{z} = \frac{z^2 + 1}{z} = \frac{7}{z} \]
From the quadratic equation, we know \( z^2 = 7z - 1 \).

### Step 8: Final calculation
Substituting back, we find:
\[ \frac{x^{72} + 1}{x^{36}} = 322 \]

Thus, the value of \( \frac{x^{72} + 1}{x^{36}} \) is:
\[ \boxed{322} \]