If `x=9−4sqrt5` then , find the value of `[x^3− 1/(x^3)]`

Options

A. `-2684sqrt(5)`
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B. `-2584sqrt(5)`
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C. `-2584sqrt(6)`
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D. `-2684sqrt(6)`
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Answer

To find the value of \( x^3 - \frac{1}{x^3} \) given \( x = 9 - 4\sqrt{5} \), we can follow these steps:

### Step 1: Find \( \frac{1}{x} \)

We start by calculating \( \frac{1}{x} \):

\[ \frac{1}{x} = \frac{1}{9 - 4\sqrt{5}} \]

To rationalize the denominator, we multiply the numerator and denominator by the conjugate \( 9 + 4\sqrt{5} \):

\[ \frac{1}{x} = \frac{1 \cdot (9 + 4\sqrt{5})}{(9 - 4\sqrt{5})(9 + 4\sqrt{5})} \]

### Step 2: Simplify the Denominator

Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):

\[ (9 - 4\sqrt{5})(9 + 4\sqrt{5}) = 9^2 - (4\sqrt{5})^2 = 81 - 80 = 1 \]

Thus, we have:

\[ \frac{1}{x} = 9 + 4\sqrt{5} \]

### Step 3: Calculate \( x + \frac{1}{x} \)

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

\[ x + \frac{1}{x} = (9 - 4\sqrt{5}) + (9 + 4\sqrt{5}) = 18 \]

### Step 4: Calculate \( x - \frac{1}{x} \)

Next, we calculate \( x - \frac{1}{x} \):

\[ x - \frac{1}{x} = (9 - 4\sqrt{5}) - (9 + 4\sqrt{5}) = -8\sqrt{5} \]

### Step 5: Use the Formula for \( x^3 - \frac{1}{x^3} \)

We can use the formula:

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

First, we need to find \( x^2 + \frac{1}{x^2} \):

\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 = 18^2 - 2 = 324 - 2 = 322 \]

### Step 6: Substitute Values into the Formula

Now substituting back into the formula:

\[ x^3 - \frac{1}{x^3} = \left( -8\sqrt{5} \right) \left( 322 + 1 \right) = -8\sqrt{5} \cdot 323 \]

### Step 7: Final Calculation

Calculating the final value:

\[ x^3 - \frac{1}{x^3} = -2584\sqrt{5} \]

Thus, the final answer is:

\[ \boxed{-2584\sqrt{5}} \]