To solve \( \sqrt{2 \times \left( \frac{1}{7} \right)^2 + 1} \), let's break it down step by step:

1. First, square \( \frac{1}{7} \):
   \[
   \left( \frac{1}{7} \right)^2 = \frac{1}{49}
   \]

2. Multiply by 2:
   \[
   2 \times \frac{1}{49} = \frac{2}{49}
   \]

3. Add 1 to the result:
   \[
   \frac{2}{49} + 1 = \frac{2}{49} + \frac{49}{49} = \frac{51}{49}
   \]

4. Now, take the square root of \( \frac{51}{49} \):
   \[
   \sqrt{\frac{51}{49}} = \frac{\sqrt{51}}{7}
   \]

Thus, the final answer is:
\[
\frac{\sqrt{51}}{7}
\]

Would you like more details on any step, or do you have any other questions?

Here are some related questions:
1. What is the square root of a fraction in general?
2. How can we simplify a square root of a non-perfect square?
3. What are some properties of square roots?
4. How does the order of operations affect this expression?
5. How does squaring and taking square roots interact?

**Tip:** Remember, when dealing with fractions, squaring or taking square roots applies to both the numerator and the denominator separately.

Learn how to solve the expression involving the square root of 2(⅐)² + 1 with a step-by-step breakdown. This problem covers key mathematical concepts like squaring fractions, applying square roots, and understanding the order of operations. Suitable for students in Grades 6-8.

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