Given: `a^(2) + (1)/(a^(2))= 7 and a ne 0`, find
`a^(2)- (1)/(a^(2))`

Answer

To solve the problem step by step, let's start with the given information and work through the calculations systematically.

### Step 1: Given Information
We are given:
\[ a^2 + \frac{1}{a^2} = 7 \]

### Step 2: Square the Given Equation
We will square both sides of the equation:
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 7^2 \]
This expands to:
\[ a^4 + 2 \cdot a^2 \cdot \frac{1}{a^2} + \frac{1}{a^4} = 49 \]
Since \( a^2 \cdot \frac{1}{a^2} = 1 \), we can simplify this to:
\[ a^4 + 2 + \frac{1}{a^4} = 49 \]

### Step 3: Rearranging 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: Finding \( a^2 - \frac{1}{a^2} \)
Next, we need to find \( a^2 - \frac{1}{a^2} \). We will use the identity:
\[ \left(a^2 - \frac{1}{a^2}\right)^2 = a^4 - 2 \cdot a^2 \cdot \frac{1}{a^2} + \frac{1}{a^4} \]
This simplifies to:
\[ \left(a^2 - \frac{1}{a^2}\right)^2 = a^4 - 2 + \frac{1}{a^4} \]

### Step 5: Substitute the Known Values
Now, substitute \( a^4 + \frac{1}{a^4} = 47 \) into the equation:
\[ \left(a^2 - \frac{1}{a^2}\right)^2 = 47 - 2 \]
\[ \left(a^2 - \frac{1}{a^2}\right)^2 = 45 \]

### Step 6: Taking the Square Root
To find \( a^2 - \frac{1}{a^2} \), we take the square root of both sides:
\[ a^2 - \frac{1}{a^2} = \pm \sqrt{45} \]
\[ a^2 - \frac{1}{a^2} = \pm 3\sqrt{5} \]

### Final Answer
Thus, the final result is:
\[ a^2 - \frac{1}{a^2} = 3\sqrt{5} \text{ or } -3\sqrt{5} \]

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