If `(x+1/x)=5` , what is the value of `(x^5 + 1/x^5)` ?

Options

A. 1875
B. 2525
C. 2530
D. 3120

Answer

To solve the problem, we need to find the value of \( x^5 + \frac{1}{x^5} \) given that \( x + \frac{1}{x} = 5 \).

### Step 1: Square the given equation
We start with the equation:
\[ x + \frac{1}{x} = 5 \]
Now, we square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 5^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 25 \]
Thus,
\[ x^2 + \frac{1}{x^2} = 25 - 2 = 23 \]

### Step 2: Cube the original equation
Next, we need to find \( x^3 + \frac{1}{x^3} \). We use the identity:
\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)\left(x^2 + \frac{1}{x^2}\right) - \left(x + \frac{1}{x}\right) \]
Substituting the known values:
\[ x^3 + \frac{1}{x^3} = 5 \cdot 23 - 5 \]
Calculating this gives:
\[ x^3 + \frac{1}{x^3} = 115 - 5 = 110 \]

### Step 3: Use the identity to find \( x^5 + \frac{1}{x^5} \)
Now we can find \( x^5 + \frac{1}{x^5} \) using the identity:
\[ x^5 + \frac{1}{x^5} = \left(x^3 + \frac{1}{x^3}\right)\left(x^2 + \frac{1}{x^2}\right) - \left(x + \frac{1}{x}\right) \]
Substituting the values we have:
\[ x^5 + \frac{1}{x^5} = 110 \cdot 23 - 5 \]
Calculating this gives:
\[ x^5 + \frac{1}{x^5} = 2530 - 5 = 2525 \]

### Final Answer
Thus, the value of \( x^5 + \frac{1}{x^5} \) is:
\[ \boxed{2525} \]