By using sultable identity, evaluate `x ^(2) + (1)/( x ^(2)).` If `x + (1)/(x) = 5.`

Answer

To evaluate \( x^2 + \frac{1}{x^2} \) given that \( x + \frac{1}{x} = 5 \), we can use the algebraic identity:

\[ \left( a + b \right)^2 = a^2 + b^2 + 2ab \]

### Step-by-Step Solution:

1. **Identify the variables**: Let \( a = x \) and \( b = \frac{1}{x} \). Thus, we have:
\[ a + b = x + \frac{1}{x} = 5 \]

2. **Square both sides**: We will square the equation \( a + b = 5 \):
\[ (a + b)^2 = 5^2 \]
This gives us:
\[ a^2 + b^2 + 2ab = 25 \]

3. **Substitute \( a^2 \) and \( b^2 \)**: We know that:
\[ a^2 = x^2 \quad \text{and} \quad b^2 = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \]
Therefore, we can rewrite the equation as:
\[ x^2 + \frac{1}{x^2} + 2ab = 25 \]

4. **Calculate \( 2ab \)**: Since \( ab = x \cdot \frac{1}{x} = 1 \), we have:
\[ 2ab = 2 \cdot 1 = 2 \]

5. **Substitute \( 2ab \) back into the equation**:
\[ x^2 + \frac{1}{x^2} + 2 = 25 \]

6. **Isolate \( x^2 + \frac{1}{x^2} \)**: To find \( x^2 + \frac{1}{x^2} \), we subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} = 25 - 2 \]
Thus, we get:
\[ x^2 + \frac{1}{x^2} = 23 \]

### Final Answer:
\[ x^2 + \frac{1}{x^2} = 23 \]