Which value would lie between the two points on the number line below?


 

A.2–√+22+2

B.53–√53

C.195√21952

D.5–√−15−1


To determine which value lies between two points on a number line, we need to evaluate the expressions provided in the answer choices and see which one falls between two specific values.

Let's evaluate each option step-by-step:

### Option A: \( \sqrt{2} + 2 \)

- \( \sqrt{2} \approx 1.414 \)
- So, \( \sqrt{2} + 2 \approx 1.414 + 2 = 3.414 \)

### Option B: \( \frac{5}{\sqrt{3}} \)

- \( \sqrt{3} \approx 1.732 \)
- So, \( \frac{5}{\sqrt{3}} \approx \frac{5}{1.732} \approx 2.887 \)

### Option C: \( \frac{19}{5\sqrt{21}} \)

- \( \sqrt{21} \approx 4.583 \)
- So, \( 5\sqrt{21} \approx 5 \times 4.583 = 22.915 \)
- \( \frac{19}{5\sqrt{21}} \approx \frac{19}{22.915} \approx 0.829 \)

### Option D: \( \sqrt{5} - 1 \)

- \( \sqrt{5} \approx 2.236 \)
- So, \( \sqrt{5} - 1 \approx 2.236 - 1 = 1.236 \)

### Summary of Evaluations

- Option A: \( \sqrt{2} + 2 \approx 3.414 \)
- Option B: \( \frac{5}{\sqrt{3}} \approx 2.887 \)
- Option C: \( \frac{19}{5\sqrt{21}} \approx 0.829 \)
- Option D: \( \sqrt{5} - 1 \approx 1.236 \)

### Conclusion

To find which value lies between the two others on a number line, we should order these values from smallest to largest:

1. Option C: \( \frac{19}{5\sqrt{21}} \approx 0.829 \)
2. Option D: \( \sqrt{5} - 1 \approx 1.236 \)
3. Option B: \( \frac{5}{\sqrt{3}} \approx 2.887 \)
4. Option A: \( \sqrt{2} + 2 \approx 3.414 \)

The value between two other values is **Option D: \( \sqrt{5} - 1 \)**.

Would you like more details on any of these calculations or have any other questions?

Here are 5 related questions for further exploration:

1. How do we rationalize the denominator in expressions like Option B?
2. How can we approximate square roots without a calculator?
3. Why is it important to compare numerical values on a number line?
4. What are the properties of square roots that help simplify expressions?
5. How do different operations (addition, subtraction, division) affect the placement of numbers on a number line?

**Tip:** When approximating square roots, remember that they are non-repeating and non-terminating decimals, making rough estimations helpful for quick calculations.

Explore how to determine which value lies between two points on a number line through a detailed analysis of square root expressions. This problem involves evaluating and comparing numerical values using square root approximations. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation