If `p + (1)/(p)=sqrt(10)`, then find the value of `p^(4) + (1)/(p^(4))`.

Options

A. 52
B. 60
C. 62
D. 65

Answer

To solve the problem step by step, we need to find the value of \( p^4 + \frac{1}{p^4} \) given that \( p + \frac{1}{p} = \sqrt{10} \).

### Step 1: Square the given equation
We start with the equation:
\[ p + \frac{1}{p} = \sqrt{10} \]
Now, we square both sides:
\[ \left(p + \frac{1}{p}\right)^2 = (\sqrt{10})^2 \]
This gives us:
\[ p^2 + 2 + \frac{1}{p^2} = 10 \]

### Step 2: Rearrange to find \( p^2 + \frac{1}{p^2} \)
From the equation above, we can rearrange it to isolate \( p^2 + \frac{1}{p^2} \):
\[ p^2 + \frac{1}{p^2} = 10 - 2 = 8 \]

### Step 3: Square the result from Step 2
Now, we need to find \( p^4 + \frac{1}{p^4} \). We can do this by squaring \( p^2 + \frac{1}{p^2} \):
\[ \left(p^2 + \frac{1}{p^2}\right)^2 = 8^2 \]
This expands to:
\[ p^4 + 2 + \frac{1}{p^4} = 64 \]

### Step 4: Rearrange to find \( p^4 + \frac{1}{p^4} \)
Now, we can rearrange this equation to isolate \( p^4 + \frac{1}{p^4} \):
\[ p^4 + \frac{1}{p^4} = 64 - 2 = 62 \]

### Final Answer
Thus, the value of \( p^4 + \frac{1}{p^4} \) is:
\[ \boxed{62} \]