If (x -1/x) = 5 , find `(x^(6) -5x^(3) -1) //(x^(6) +7x^(3) -1)`

Options

A. (A). ` 145 //133`
B. (B). ` 135//147`
C. (C). ` 133//145`
D. (D). ` 135//133`

Answer

To solve the problem, we start with the equation given:

\[ x - \frac{1}{x} = 5 \]

### Step 1: Find \( x^3 - \frac{1}{x^3} \)

To find \( x^3 - \frac{1}{x^3} \), we can use the identity:

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

Substituting the value we have:

\[ x^3 - \frac{1}{x^3} = 5^3 + 3 \cdot 5 \]

Calculating \( 5^3 \):

\[ 5^3 = 125 \]

Calculating \( 3 \cdot 5 \):

\[ 3 \cdot 5 = 15 \]

Now adding these values together:

\[ x^3 - \frac{1}{x^3} = 125 + 15 = 140 \]

### Step 2: Substitute into the original expression

We need to find:

\[ \frac{x^6 - 5x^3 - 1}{x^6 + 7x^3 - 1} \]

### Step 3: Express \( x^6 \) in terms of \( x^3 \)

We can express \( x^6 \) as:

\[ x^6 = \left( x^3 \right)^2 \]

Thus, we can rewrite the expression as:

\[ \frac{(x^3)^2 - 5x^3 - 1}{(x^3)^2 + 7x^3 - 1} \]

Let \( y = x^3 \). The expression simplifies to:

\[ \frac{y^2 - 5y - 1}{y^2 + 7y - 1} \]

### Step 4: Substitute \( y = x^3 \)

We know \( x^3 - \frac{1}{x^3} = 140 \), which means:

\[ y - \frac{1}{y} = 140 \]

From this, we can find \( y + \frac{1}{y} \):

\[ y + \frac{1}{y} = \frac{(y - \frac{1}{y})^2 + 4}{(y - \frac{1}{y})} = \frac{140^2 + 4}{140} \]

Calculating \( 140^2 \):

\[ 140^2 = 19600 \]

So:

\[ y + \frac{1}{y} = \frac{19600 + 4}{140} = \frac{19604}{140} = 140.02857 \approx 140 \]

### Step 5: Evaluate the expression

Now we can evaluate the expression:

\[ \frac{y^2 - 5y - 1}{y^2 + 7y - 1} \]

Substituting \( y = 140 \):

\[ \frac{140^2 - 5 \cdot 140 - 1}{140^2 + 7 \cdot 140 - 1} \]

Calculating \( 140^2 - 5 \cdot 140 - 1 \):

\[ 140^2 = 19600 \]
\[ 5 \cdot 140 = 700 \]
\[ 19600 - 700 - 1 = 18899 \]

Calculating \( 140^2 + 7 \cdot 140 - 1 \):

\[ 7 \cdot 140 = 980 \]
\[ 19600 + 980 - 1 = 20579 \]

Finally, the expression becomes:

\[ \frac{18899}{20579} \]

### Final Answer

Thus, the final answer is:

\[ \frac{18899}{20579} \]