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To compare \( -\sqrt{7} \) and \( -3.12345\ldots \), we will calculate the approximate value of \( \sqrt{7} \) and then compare the two numbers.

1. \( \sqrt{7} \approx 2.64575 \), so:
   \[
   -\sqrt{7} \approx -2.64575
   \]

2. The second number is \( -3.12345\ldots \), which is already given.

Now let's compare:
- \( -\sqrt{7} \approx -2.64575 \)
- \( -3.12345\ldots \)

Since \( -3.12345\ldots \) is smaller (more negative) than \( -2.64575 \), we can conclude that:
\[
-\sqrt{7} > -3.12345\ldots
\]

### Justification:
The comparison is based on their decimal approximations. Since \( -\sqrt{7} \) is closer to 0 than \( -3.12345\ldots \), it is the larger of the two numbers.

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Would you like further details on how the square root was calculated or any other questions?

### 5 Related Questions:
1. How do you approximate square roots without a calculator?
2. What is the exact value of \( \sqrt{7} \)?
3. Why is \( -2.64575 \) larger than \( -3.12345 \)?
4. How can you compare square roots of non-perfect squares?
5. What other methods can be used to compare irrational and decimal numbers?

### Tip:
When comparing negative numbers, the one closer to zero is always the larger number.

Compare -√7 and -3.12345.... Justify your reasoning.

Learn how to compare -√7 and -3.12345 with a detailed step-by-step explanation. This problem covers key mathematical concepts including square roots, irrational numbers, and negative number comparison. Suitable for students in Grades 9-12.

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