If `a-(1)/(a)=7`, then `a^(2)+(1)/(a^(2))=`?

Options

A. 54
B. 51
C. 53
D. 52

Answer

To solve the equation \( a - \frac{1}{a} = 7 \) and find the value of \( a^2 + \frac{1}{a^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ a - \frac{1}{a} = 7 \]
Now, we square both sides:
\[ \left(a - \frac{1}{a}\right)^2 = 7^2 \]
This simplifies to:
\[ a^2 - 2a \cdot \frac{1}{a} + \frac{1}{a^2} = 49 \]

### Step 2: Simplify the equation
The term \( -2a \cdot \frac{1}{a} \) simplifies to \( -2 \):
\[ a^2 - 2 + \frac{1}{a^2} = 49 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} = 49 + 2 \]
This simplifies to:
\[ a^2 + \frac{1}{a^2} = 51 \]

### Final Answer
Thus, the value of \( a^2 + \frac{1}{a^2} \) is:
\[ \boxed{51} \]