Let's go through each expression one-by-one and match them with their estimated values:
1. Expression A: [tex]\(\sqrt{90} - \sqrt{40}\)[/tex]
- The numerical value for this expression is approximately [tex]\(3.162277660168379\)[/tex].
2. Expression B: [tex]\(\frac{\sqrt{35} - \sqrt{42}}{\sqrt{5} - \sqrt{6}}\)[/tex]
- The numerical value for this expression is approximately [tex]\(2.6457513110645947\)[/tex].
3. Expression C: [tex]\(2 \sqrt{27} - \sqrt{48}\)[/tex]
- The numerical value for this expression is approximately [tex]\(3.4641016151377553\)[/tex].
4. Expression D: [tex]\(\frac{\sqrt{54} - \sqrt{24}}{\sqrt{18} - \sqrt{8}}\)[/tex]
- The numerical value for this expression is approximately [tex]\(1.7320508075688785\)[/tex].
Now we will match each value with the given expressions and their positions on the number line:
- For [tex]\(\sqrt{90} - \sqrt{40}\)[/tex]: Its value is [tex]\(3.162277660168379\)[/tex]. Hence, it is located around 3.162 on the number line.
- For [tex]\(\frac{\sqrt{35} - \sqrt{42}}{\sqrt{5} - \sqrt{6}}\)[/tex]: Its value is [tex]\(2.6457513110645947\)[/tex]. This places it around 2.646 on the number line.
- For [tex]\(2 \sqrt{27} - \sqrt{48}\)[/tex]: Its value is [tex]\(3.4641016151377553\)[/tex]. Therefore, it is around 3.464 on the number line.
- For [tex]\(\frac{\sqrt{54} - \sqrt{24}}{\sqrt{18} - \sqrt{8}}\)[/tex]: Its value is [tex]\(1.7320508075688785\)[/tex]. Hence, it is located around 1.732 on the number line.
Matching these values to the given expressions:
- [tex]\( \sqrt{90} - \sqrt{40} \approx 3.162 \)[/tex]
- [tex]\( \frac{\sqrt{35} - \sqrt{42}}{\sqrt{5} - \sqrt{6}} \approx 2.646 \)[/tex]
- [tex]\( 2 \sqrt{27} - \sqrt{48} \approx 3.464 \)[/tex]
- [tex]\( \frac{\sqrt{54} - \sqrt{24}}{\sqrt{18} - \sqrt{8}} \approx 1.732 \)[/tex]
This matches the values [tex]\(3.162277660168379\)[/tex], [tex]\(2.6457513110645947\)[/tex], [tex]\(3.4641016151377553\)[/tex], and [tex]\(1.7320508075688785\)[/tex] with the provided expressions accordingly.
