If `a+1/a=4`. then the value of `a^2+1/a^2` is :

Options

A. 6
B. 7
C. 9
D. can't be determined

Answer

To solve the problem where \( a + \frac{1}{a} = 4 \) and we need to 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:
\[ a + \frac{1}{a} = 4 \]
Now, we square both sides:
\[ \left( a + \frac{1}{a} \right)^2 = 4^2 \]

### Step 2: Apply the square of a binomial formula
Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \), we can expand the left side:
\[ a^2 + 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 16 \]
Since \( a \cdot \frac{1}{a} = 1 \), we simplify this to:
\[ a^2 + 2 + \frac{1}{a^2} = 16 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} + 2 = 16 \]
Subtract 2 from both sides:
\[ a^2 + \frac{1}{a^2} = 16 - 2 \]

### Step 4: Simplify the result
Now we simplify the right side:
\[ a^2 + \frac{1}{a^2} = 14 \]

### Final Answer
Thus, the value of \( a^2 + \frac{1}{a^2} \) is:
\[ \boxed{14} \]