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

Options

A. 433
B. 322
C. 345
D. 432

Answer

To solve the problem, we start with the given equation:

\[ \frac{x^{24} + 1}{x^{12}} = 7 \]

### Step 1: Rewrite the equation
We can rewrite the equation as follows:

\[ \frac{x^{24}}{x^{12}} + \frac{1}{x^{12}} = 7 \]

This simplifies to:

\[ x^{12} + \frac{1}{x^{12}} = 7 \]

### Step 2: Let \( y = x^{12} \)
Now, let’s set \( y = x^{12} \). The equation becomes:

\[ y + \frac{1}{y} = 7 \]

### Step 3: Multiply both sides by \( y \)
To eliminate the fraction, multiply both sides by \( y \):

\[ y^2 + 1 = 7y \]

### Step 4: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:

\[ y^2 - 7y + 1 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1 \), \( b = -7 \), and \( c = 1 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot 1 = 49 - 4 = 45 \]

Now substituting into the quadratic formula:

\[ y = \frac{7 \pm \sqrt{45}}{2} \]

Since \( \sqrt{45} = 3\sqrt{5} \), we have:

\[ y = \frac{7 \pm 3\sqrt{5}}{2} \]

### Step 6: Calculate \( x^{36} + \frac{1}{x^{36}} \)
Next, we need to find \( \frac{x^{72} + 1}{x^{36}} \). First, we find \( x^{36} + \frac{1}{x^{36}} \).

Using the identity:

\[ \left( y + \frac{1}{y} \right)^2 = y^2 + 2 + \frac{1}{y^2} \]

We can find \( x^{24} + \frac{1}{x^{24}} \):

\[ \left( 7 \right)^2 = y^2 + 2 + \frac{1}{y^2} \]

This gives:

\[ 49 = y^2 + 2 + \frac{1}{y^2} \implies y^2 + \frac{1}{y^2} = 49 - 2 = 47 \]

Now, we can find \( x^{36} + \frac{1}{x^{36}} \):

Using the identity again:

\[ \left( y^2 + \frac{1}{y^2} \right) \left( y + \frac{1}{y} \right) = y^3 + y + \frac{1}{y} + \frac{1}{y^3} \]

Thus:

\[ x^{36} + \frac{1}{x^{36}} = 47 \cdot 7 - 7 = 329 - 7 = 322 \]

### Step 7: Find \( \frac{x^{72} + 1}{x^{36}} \)
Now we can calculate:

\[ \frac{x^{72} + 1}{x^{36}} = x^{36} + \frac{1}{x^{36}} = 322 \]

### Final Answer
Thus, the value of \( \frac{x^{72} + 1}{x^{36}} \) is:

\[ \boxed{322} \]