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

Options

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

Answer

To solve the problem, we will follow these steps:

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

**Step 1: Simplify the given equation.**
We can rewrite the equation as:
\[ x^{24} + 1 = 7x^{12} \]

**Step 2: Rearranging the equation.**
Now, we can rearrange this to:
\[ x^{24} - 7x^{12} + 1 = 0 \]

**Step 3: Let \( y = x^{12} \).**
Substituting \( y \) into the equation gives us:
\[ y^2 - 7y + 1 = 0 \]

**Step 4: Solve the quadratic equation.**
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -7, c = 1 \):
\[ y = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ y = \frac{7 \pm \sqrt{49 - 4}}{2} \]
\[ y = \frac{7 \pm \sqrt{45}}{2} \]
\[ y = \frac{7 \pm 3\sqrt{5}}{2} \]

**Step 5: Find \( x^{36} + \frac{1}{x^{36}} \).**
We need to find:
\[ \frac{x^{72} + 1}{x^{36}} = x^{36} + \frac{1}{x^{36}} \]
Using the identity \( a^3 + \frac{1}{a^3} = (a + \frac{1}{a})^3 - 3(a + \frac{1}{a}) \), we can find \( x^{36} + \frac{1}{x^{36}} \).

**Step 6: Calculate \( x^{36} + \frac{1}{x^{36}} \).**
Let \( a = x^{12} \), then:
\[ x^{36} = (x^{12})^3 = a^3 \]
We know \( a + \frac{1}{a} = 7 \). Therefore:
\[ a^3 + \frac{1}{a^3} = (a + \frac{1}{a})^3 - 3(a + \frac{1}{a}) \]
Substituting \( a + \frac{1}{a} = 7 \):
\[ a^3 + \frac{1}{a^3} = 7^3 - 3 \cdot 7 \]
\[ = 343 - 21 = 322 \]

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