if x + 1/x = squareroot of 71, what is the value of x^2 + 1/x^2

We are given:

\[
x + \frac{1}{x} = \sqrt{71}
\]

We need to find the value of \( x^2 + \frac{1}{x^2} \).

### Step-by-Step Process:

1. **Square both sides of the equation** to relate \( x^2 + \frac{1}{x^2} \) to \( x + \frac{1}{x} \).

   Squaring \( x + \frac{1}{x} \):

   \[
   \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2}
   \]

   Therefore:

   \[
   x^2 + \frac{1}{x^2} + 2 = \left( \sqrt{71} \right)^2
   \]

2. **Simplify** the equation:

   \[
   x^2 + \frac{1}{x^2} + 2 = 71
   \]

   Subtract 2 from both sides:

   \[
   x^2 + \frac{1}{x^2} = 71 - 2
   \]

   \[
   x^2 + \frac{1}{x^2} = 69
   \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{69} \).

Would you like more details or have any questions?

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Here are 5 related questions for deeper understanding:
1. If \( x + \frac{1}{x} = 5 \), what is \( x^2 + \frac{1}{x^2} \)?
2. If \( x + \frac{1}{x} = 10 \), what is the value of \( x^2 + \frac{1}{x^2} \)?
3. How would you solve \( x^2 + \frac{1}{x^2} = 45 \) for \( x + \frac{1}{x} \)?
4. Can you express \( x^3 + \frac{1}{x^3} \) in terms of \( x + \frac{1}{x} \)?
5. What would happen if \( x + \frac{1}{x} = 0 \)? What is \( x^2 + \frac{1}{x^2} \)?

**Tip**: Always remember that squaring both sides of an equation can introduce extraneous solutions, so it's good to verify the solutions afterward if possible.

If x + 1/x = √71, what is the value of x^2 + 1/x^2?

Learn how to find the value of x^2 + 1/x^2 when x + 1/x = √71. This problem involves algebraic identities, binomial expansion, and simplification techniques. Suitable for students in Grades 9-11.

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