If `(x^(2)+1/(x^(2)))=34,(x gt0)` , then find the value of `x^(3)+1/(x^(3))`

Options

A. 198
B. 216
C. 200
D. 186

Answer

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

1. **Given Equation**:
\[ x^2 + \frac{1}{x^2} = 34 \]

2. **Finding \( x + \frac{1}{x} \)**:
We can use the identity:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Rearranging this gives:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting the given value:
\[ 34 = \left( x + \frac{1}{x} \right)^2 - 2 \]
Adding 2 to both sides:
\[ \left( x + \frac{1}{x} \right)^2 = 36 \]
Taking the square root:
\[ x + \frac{1}{x} = 6 \quad (\text{since } x > 0) \]

3. **Finding \( x^3 + \frac{1}{x^3} \)**:
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 \( x + \frac{1}{x} = 6 \):
\[ x^3 + \frac{1}{x^3} = 6^3 - 3 \cdot 6 \]
Calculating \( 6^3 \):
\[ 6^3 = 216 \]
Calculating \( 3 \cdot 6 \):
\[ 3 \cdot 6 = 18 \]
Thus:
\[ x^3 + \frac{1}{x^3} = 216 - 18 = 198 \]

4. **Final Answer**:
\[ x^3 + \frac{1}{x^3} = 198 \]