If ` x^( 2) + (1)/( 25 x ^(2)) = (8)/( 5) and x gt 0 ` then what is the value of ` ( x^(3) + (1)/( 125 x^(3))) ` ?

Options

A. ` 7 sqrt(2)`
B. ` 5 sqrt(2)`
C. ` (( 7 sqrt(2)))/(5)`
D. ` 7 sqrt(6)`

Answer

To solve the equation \( x^2 + \frac{1}{25x^2} = \frac{8}{5} \) and find the value of \( x^3 + \frac{1}{125x^3} \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ x^2 + \frac{1}{25x^2} = \frac{8}{5} \]

### Step 2: Introduce a new variable
Let \( y = x + \frac{1}{5x} \). Then, we can express \( x^2 + \frac{1}{25x^2} \) in terms of \( y \):
\[ y^2 = \left( x + \frac{1}{5x} \right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{5x} + \frac{1}{25x^2} = x^2 + \frac{1}{25x^2} + \frac{2}{5} \]
Thus, we can rewrite:
\[ x^2 + \frac{1}{25x^2} = y^2 - \frac{2}{5} \]

### Step 3: Substitute into the equation
Now substitute this into the original equation:
\[ y^2 - \frac{2}{5} = \frac{8}{5} \]
Adding \( \frac{2}{5} \) to both sides gives:
\[ y^2 = \frac{8}{5} + \frac{2}{5} = \frac{10}{5} = 2 \]

### Step 4: Solve for \( y \)
Taking the square root of both sides, we find:
\[ y = \sqrt{2} \]
Since \( x > 0 \), we take the positive root.

### Step 5: Find \( x^3 + \frac{1}{125x^3} \)
We know:
\[ x^3 + \frac{1}{125x^3} = \left( x + \frac{1}{5x} \right)^3 - 3 \left( x \cdot \frac{1}{5x} \right) \left( x + \frac{1}{5x} \right) \]
This simplifies to:
\[ x^3 + \frac{1}{125x^3} = y^3 - 3 \cdot \frac{1}{5} \cdot y \]
Substituting \( y = \sqrt{2} \):
\[ x^3 + \frac{1}{125x^3} = (\sqrt{2})^3 - 3 \cdot \frac{1}{5} \cdot \sqrt{2} \]
Calculating \( (\sqrt{2})^3 \):
\[ (\sqrt{2})^3 = 2\sqrt{2} \]
So we have:
\[ x^3 + \frac{1}{125x^3} = 2\sqrt{2} - \frac{3\sqrt{2}}{5} \]

### Step 6: Combine the terms
To combine the terms, we need a common denominator:
\[ 2\sqrt{2} = \frac{10\sqrt{2}}{5} \]
Thus:
\[ x^3 + \frac{1}{125x^3} = \frac{10\sqrt{2}}{5} - \frac{3\sqrt{2}}{5} = \frac{7\sqrt{2}}{5} \]

### Final Answer
The value of \( x^3 + \frac{1}{125x^3} \) is:
\[ \frac{7\sqrt{2}}{5} \]