Certainly! Let's tackle this step by step and match the correct expressions with their respective positions on the number line.
First, let's summarize the values for each of the expressions as follows:
- Expression A: [tex]\(\sqrt{90} - \sqrt{40}\)[/tex] has an approximate value of 3.16.
- Expression B: [tex]\(\frac{\sqrt{25} - \sqrt{42}}{\sqrt{5} - \sqrt{6}}\)[/tex] has an approximate value of 6.94.
- Expression C: [tex]\(2 \sqrt{27} - \sqrt{48}\)[/tex] has an approximate value of 3.46.
- Expression D: [tex]\(\frac{\sqrt{54} - \sqrt{24}}{\sqrt{18} - \sqrt{8}}\)[/tex] has an approximate value of 1.73.
Let's match each expression with its corresponding letter from the question:
1. [tex]\(\sqrt{90} - \sqrt{40}\)[/tex] corresponds to the value 3.16, which is Expression A.
2. [tex]\(\frac{\sqrt{25} - \sqrt{42}}{\sqrt{5} - \sqrt{6}}\)[/tex] corresponds to the value 6.94, which is Expression B.
3. [tex]\(2 \sqrt{27} - \sqrt{48}\)[/tex] corresponds to the value 3.46, which is Expression C.
4. [tex]\(\frac{\sqrt{54} - \sqrt{24}}{\sqrt{18} - \sqrt{8}}\)[/tex] corresponds to the value 1.73, which is Expression D.
Final matching:
- [tex]\(\sqrt{90} - \sqrt{40}\)[/tex] : A (3.16)
- [tex]\(\frac{\sqrt{25} - \sqrt{42}}{\sqrt{5} - \sqrt{6}}\)[/tex] : B (6.94)
- [tex]\(2 \sqrt{27} - \sqrt{48}\)[/tex] : C (3.46)
- [tex]\(\frac{\sqrt{54} - \sqrt{24}}{\sqrt{18} - \sqrt{8}}\)[/tex] : D (1.73)
This step-by-step matching ensures that each expression is correctly paired with its value, allowing us to accurately place them on the number line.
