Question

Compare √42 and √58 plotted on the number line. What is the approximate difference in tenths between the two values?

A number line going from 6 to 10. StartRoot 42 EndRoot is plotted between 6 and 7, and StartRoot 58 EndRoot is plotted between 7 and 8.

StartRoot 58 EndRoot is about
greater than StartRoot 42 EndRoot.

Answer 1

To compare \( \sqrt{42} \) and \( \sqrt{58} \) and find the approximate difference between the two:

1. **Estimate \( \sqrt{42} \)**:
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
- Therefore, \( 6 < \sqrt{42} < 7 \).
- We can refine our estimation: \( 6.4^2 = 40.96 \) and \( 6.5^2 = 42.25 \). Therefore, \( \sqrt{42} \approx 6.5 \).

2. **Estimate \( \sqrt{58} \)**:
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
- Therefore, \( 7 < \sqrt{58} < 8 \).
- We can refine our estimation: \( 7.5^2 = 56.25 \) and \( 7.6^2 = 57.76 \). Therefore, \( \sqrt{58} \approx 7.6 \).

3. **Calculate the difference**:
- Approximate values:
- \( \sqrt{42} \approx 6.5 \)
- \( \sqrt{58} \approx 7.6 \)
- Difference:
\[
7.6 - 6.5 = 1.1
\]

Thus, \( \sqrt{58} \) is approximately 1.1 greater than \( \sqrt{42} \).

So, the answer is:

**\( \sqrt{58} \) is about 1.1 greater than \( \sqrt{42} \).**