If `x + (1)/(16 x)` =1 then the value of `64 x ^(3) + (1)/( 64 x ^(3))` is

Options

A. 4
B. 52
C. 64
D. 76

Answer

To solve the equation \( x + \frac{1}{16x} = 1 \) and find the value of \( 64x^3 + \frac{1}{64x^3} \), we can follow these steps:

### Step 1: Simplify the given equation
Starting with the equation:
\[ x + \frac{1}{16x} = 1 \]
We can multiply both sides by \( 16x \) to eliminate the fraction:
\[ 16x^2 + 1 = 16x \]

### Step 2: Rearrange the equation
Now, rearranging the equation gives:
\[ 16x^2 - 16x + 1 = 0 \]

### Step 3: Apply the quadratic formula
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 16, b = -16, c = 1 \):
\[ b^2 - 4ac = (-16)^2 - 4 \cdot 16 \cdot 1 = 256 - 64 = 192 \]
So,
\[ x = \frac{16 \pm \sqrt{192}}{32} \]
\[ \sqrt{192} = 8\sqrt{3} \quad \text{(since } 192 = 64 \cdot 3\text{)} \]
Thus,
\[ x = \frac{16 \pm 8\sqrt{3}}{32} = \frac{2 \pm \sqrt{3}}{4} \]

### Step 4: Find \( 4x + \frac{1}{4x} \)
Next, we calculate \( 4x + \frac{1}{4x} \):
Let \( y = 4x \). Then,
\[ y + \frac{1}{4x} = 4 \cdot \frac{2 \pm \sqrt{3}}{4} + \frac{1}{4 \cdot \frac{2 \pm \sqrt{3}}{4}} = 2 \pm \sqrt{3} + \frac{1}{2 \pm \sqrt{3}} \]
To simplify \( \frac{1}{2 \pm \sqrt{3}} \), we can multiply the numerator and denominator by the conjugate:
\[ \frac{1}{2 \pm \sqrt{3}} \cdot \frac{2 \mp \sqrt{3}}{2 \mp \sqrt{3}} = \frac{2 \mp \sqrt{3}}{4 - 3} = 2 \mp \sqrt{3} \]
Thus,
\[ 4x + \frac{1}{4x} = (2 \pm \sqrt{3}) + (2 \mp \sqrt{3}) = 4 \]

### Step 5: Calculate \( 64x^3 + \frac{1}{64x^3} \)
Now we can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Let \( a = 4x \) and \( b = \frac{1}{4x} \):
\[ 64x^3 + \frac{1}{64x^3} = (4x + \frac{1}{4x})^3 - 3(4x)(\frac{1}{4x})(4x + \frac{1}{4x}) \]
Substituting \( 4x + \frac{1}{4x} = 4 \):
\[ = 4^3 - 3(1)(4) = 64 - 12 = 52 \]

### Final Answer
Thus, the value of \( 64x^3 + \frac{1}{64x^3} \) is:
\[ \boxed{52} \]