If ` x+(1)/(x) =4 , " then " x^(3) + (1)/(x^(3))` is equal to :

Options

A. 64
B. 50
C. 52
D. 40

Answer

To solve the problem, we need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x + \frac{1}{x} = 4 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + \frac{1}{x} = 4 \]

2. **Cube both sides**:
We will use the identity for the cube of a sum:
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]
Here, let \( a = x \) and \( b = \frac{1}{x} \). Therefore, we have:
\[ \left(x + \frac{1}{x}\right)^3 = x^3 + \frac{1}{x^3} + 3\left(x \cdot \frac{1}{x}\right)\left(x + \frac{1}{x}\right) \]

3. **Substituting the known values**:
From the previous step, substituting \( x + \frac{1}{x} = 4 \):
\[ 4^3 = x^3 + \frac{1}{x^3} + 3(1)(4) \]

4. **Calculate \( 4^3 \)**:
\[ 4^3 = 64 \]

5. **Substituting back into the equation**:
\[ 64 = x^3 + \frac{1}{x^3} + 12 \]

6. **Isolate \( x^3 + \frac{1}{x^3} \)**:
\[ x^3 + \frac{1}{x^3} = 64 - 12 \]
\[ x^3 + \frac{1}{x^3} = 52 \]

### Final Answer:
Thus, the value of \( x^3 + \frac{1}{x^3} \) is \( \boxed{52} \).