If `a+(1)/(a)=3`, then the value of `a^(2)+(1)/(a^(2))` is _________.

Options

A. 9
B. 6
C. 7
D. 8

Answer

To solve the equation \( a + \frac{1}{a} = 3 \) 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:
\[ a + \frac{1}{a} = 3 \]
Now, we square both sides:
\[ \left(a + \frac{1}{a}\right)^2 = 3^2 \]

### Step 2: Expand the left side using the identity
Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \), we can expand the left side:
\[ a^2 + 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 9 \]
This simplifies to:
\[ a^2 + 2 + \frac{1}{a^2} = 9 \]

### Step 3: Rearrange the equation
Now, we rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} + 2 = 9 \]
Subtract 2 from both sides:
\[ a^2 + \frac{1}{a^2} = 9 - 2 \]

### Step 4: Simplify the expression
Now, we simplify the right side:
\[ a^2 + \frac{1}{a^2} = 7 \]

### Final Answer
Thus, the value of \( a^2 + \frac{1}{a^2} \) is \( \boxed{7} \).

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