If `a^(2)+1/(a^(2))=7`, find the value of
`(a^(2)-1/(a^(2)))`

Answer

To solve the problem, we need to find the value of \( a^2 - \frac{1}{a^2} \) given that \( a^2 + \frac{1}{a^2} = 7 \).

### Step 1: Start with the given equation
We have:
\[ a^2 + \frac{1}{a^2} = 7 \]

### Step 2: Square both sides
We will square both sides of the equation to use the identity:
\[ \left( a^2 + \frac{1}{a^2} \right)^2 = 7^2 \]
This expands to:
\[ a^4 + 2 + \frac{1}{a^4} = 49 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( a^4 + \frac{1}{a^4} \):
\[ a^4 + \frac{1}{a^4} = 49 - 2 \]
\[ a^4 + \frac{1}{a^4} = 47 \]

### Step 4: Use the identity for \( a^2 - \frac{1}{a^2} \)
We know that:
\[ a^2 - \frac{1}{a^2} = \sqrt{(a^2 + \frac{1}{a^2})^2 - 4} \]
Substituting the value we have:
\[ a^2 - \frac{1}{a^2} = \sqrt{(7)^2 - 4} \]
\[ = \sqrt{49 - 4} \]
\[ = \sqrt{45} \]

### Step 5: Simplify the square root
We can simplify \( \sqrt{45} \):
\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]

### Final Answer
Thus, the value of \( a^2 - \frac{1}{a^2} \) is:
\[ \boxed{3\sqrt{5}} \]