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

Options

A. a) 50
B. b) 52
C. c) 64
D. d) 40

Answer

To solve the equation \( x + \frac{1}{x} = 4 \) and find the value of \( x^3 + \frac{1}{x^3} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x + \frac{1}{x} = 4 \]

### Step 2: Cube both sides
To find \( x^3 + \frac{1}{x^3} \), we can use the identity:
\[ \left( a + b \right)^3 = a^3 + b^3 + 3ab(a + b) \]
In our case, let \( a = x \) and \( b = \frac{1}{x} \). Therefore, we can cube both sides:
\[ \left( x + \frac{1}{x} \right)^3 = 4^3 \]
Calculating the right side:
\[ 4^3 = 64 \]

### Step 3: Expand the left side using the identity
Using the identity mentioned:
\[ x^3 + \frac{1}{x^3} + 3 \left( x \cdot \frac{1}{x} \right) \left( x + \frac{1}{x} \right) = 64 \]
Since \( x \cdot \frac{1}{x} = 1 \), we can simplify:
\[ x^3 + \frac{1}{x^3} + 3(1)(4) = 64 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 12 = 64 \]

### Step 4: Solve for \( x^3 + \frac{1}{x^3} \)
Now, we isolate \( x^3 + \frac{1}{x^3} \):
\[ x^3 + \frac{1}{x^3} = 64 - 12 \]
Calculating the right side gives:
\[ x^3 + \frac{1}{x^3} = 52 \]

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

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