If `x ^(2) + (1)/(x ^(2)) = 98 ( x gt 0),` then the valu eof `x ^(3) + (1)/( x ^(3))` is.

Options

A. 970
B. 1030
C. `-970`
D. `-1030`

Answer

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

1. **Given**:
\[ x^2 + \frac{1}{x^2} = 98 \]

2. **Step 1**: We can relate \(x^2 + \frac{1}{x^2}\) to \(x + \frac{1}{x}\) using the identity:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]
Let \(y = x + \frac{1}{x}\). Then we have:
\[ x^2 + \frac{1}{x^2} = y^2 - 2 \]

3. **Step 2**: Substitute the value of \(x^2 + \frac{1}{x^2}\) into the equation:
\[ y^2 - 2 = 98 \]

4. **Step 3**: Solve for \(y^2\):
\[ y^2 = 98 + 2 = 100 \]

5. **Step 4**: Take the square root of both sides:
\[ y = \sqrt{100} = 10 \]
(Since \(x > 0\), we take the positive root.)

6. **Step 5**: Now, we need to find \(x^3 + \frac{1}{x^3}\). We can use the identity:
\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)\left(x^2 - 1 + \frac{1}{x^2}\right) \]
We already know \(x + \frac{1}{x} = y = 10\) and we can find \(x^2 + \frac{1}{x^2} = 98\).

7. **Step 6**: Substitute into the identity:
\[ x^3 + \frac{1}{x^3} = y \left( x^2 + \frac{1}{x^2} - 1 \right) \]
\[ = 10 \left( 98 - 1 \right) = 10 \times 97 = 970 \]

8. **Final Answer**:
\[ x^3 + \frac{1}{x^3} = 970 \]