If ` (1)/( a) ( a^(2) + 1) = 3 ` then the value of ` (( a^(6) + 1)/( a^(3)))` is

Options

A. 9
B. 18
C. 27
D. 1

Answer

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

### Step 1: Simplify the given equation
Start with the equation:
\[ \frac{1}{a} (a^2 + 1) = 3 \]
Multiply both sides by \( a \) to eliminate the fraction:
\[ a^2 + 1 = 3a \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ a^2 - 3a + 1 = 0 \]

### Step 3: Use the quadratic formula
To solve for \( a \), we can use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -3, c = 1 \):
\[ a = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ a = \frac{3 \pm \sqrt{9 - 4}}{2} \]
\[ a = \frac{3 \pm \sqrt{5}}{2} \]

### Step 4: Find \( a^3 + \frac{1}{a^3} \)
Next, we need to find \( a^3 + \frac{1}{a^3} \). We can use the identity:
\[ a^3 + \frac{1}{a^3} = \left( a + \frac{1}{a} \right)^3 - 3 \left( a + \frac{1}{a} \right) \]
First, we find \( a + \frac{1}{a} \):
From the rearranged equation \( a^2 - 3a + 1 = 0 \), we can express \( a + \frac{1}{a} \):
\[ a + \frac{1}{a} = 3 \]

### Step 5: Calculate \( a^3 + \frac{1}{a^3} \)
Now substitute \( a + \frac{1}{a} \) into the identity:
\[ a^3 + \frac{1}{a^3} = 3^3 - 3 \cdot 3 \]
\[ = 27 - 9 = 18 \]

### Step 6: Find \( \frac{a^6 + 1}{a^3} \)
Now, we can find \( \frac{a^6 + 1}{a^3} \):
\[ \frac{a^6 + 1}{a^3} = a^3 + \frac{1}{a^3} \]
From our previous calculation:
\[ \frac{a^6 + 1}{a^3} = 18 \]

Thus, the value of \( \frac{a^6 + 1}{a^3} \) is \( \boxed{18} \).