If `x+(1)/(x)=4` , then find `x^(3)+(1)/(x^(3))`

Answer

To solve the equation \( x + \frac{1}{x} = 4 \) and find the value of \( x^3 + \frac{1}{x^3} \), we can use the identity for the cube of a sum. Here’s how to do it step-by-step:

### Step 1: Use the identity for cubes
We know that:
\[ a + b = x + \frac{1}{x} \]
Let \( a = x \) and \( b = \frac{1}{x} \). The identity states:
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]

### Step 2: Substitute values into the identity
From the equation \( x + \frac{1}{x} = 4 \), we can substitute this into the identity:
\[ (x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3 \cdot x \cdot \frac{1}{x} \cdot (x + \frac{1}{x}) \]

### Step 3: Simplify the equation
We know that \( x \cdot \frac{1}{x} = 1 \), so the equation becomes:
\[ 4^3 = x^3 + \frac{1}{x^3} + 3 \cdot 1 \cdot 4 \]
Calculating \( 4^3 \):
\[ 64 = x^3 + \frac{1}{x^3} + 12 \]

### Step 4: Isolate \( x^3 + \frac{1}{x^3} \)
Now, we can isolate \( x^3 + \frac{1}{x^3} \):
\[ x^3 + \frac{1}{x^3} = 64 - 12 \]

### Step 5: Calculate the final value
\[ x^3 + \frac{1}{x^3} = 52 \]

Thus, the final answer is:
\[ \boxed{52} \]