https://file.aiquickdraw.com/mathbot/file/1727280453297dql4c05f.jpg

The problem asks you to compare \(-\sqrt{7}\) and \(-3.12345\ldots\) and justify your reasoning.

### Let's break it down:

1. **Estimate \(-\sqrt{7}\)**:
   \[
   \sqrt{7} \approx 2.64575
   \]
   Therefore:
   \[
   -\sqrt{7} \approx -2.64575
   \]

2. **Comparing \(-3.12345\ldots\) and \(-\sqrt{7}\)**:
   - We know that \(-3.12345\ldots\) is smaller than \(-2.64575\) because \(-3.12345\) is further left on the number line than \(-2.64575\).
   - Thus, \(-3.12345 < -\sqrt{7}\).

### Justifying the reasoning:
- The square root of 7 is between 2 and 3, meaning \(-\sqrt{7}\) is between \(-3\) and \(-2\).
- Since \(-3.12345\) is less than \(-3\), it is less than \(-\sqrt{7}\).

### Filling in the blanks:
1. **Blank 1:** \(-\sqrt{7}\) is between \(-3\) and \(-2\).
2. **Blank 2:** Since \(-3.12345\) is **greater than** \(-3\), it is **less than** \(-\sqrt{7}\).

---

Do you need further clarification, or do you have any questions?

### Related questions:
1. How do you approximate square roots without a calculator?
2. What does it mean for a number to be greater or less than another on a number line?
3. How does comparing negative numbers differ from comparing positive numbers?
4. What are other ways to estimate irrational numbers like \(\sqrt{7}\)?
5. Can we compare numbers using fractions instead of decimals?

**Tip:** When comparing negative numbers, remember that numbers closer to zero are always greater than those further from zero!

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

This exercise involves comparing two negative real numbers: -√7 and -3.12345. Learn how to use square roots and inequalities to justify the comparison. The problem explores the concept of negative numbers and their placement on a number line. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation