Certainly! Let's start with the given surds [tex]\(10 \sqrt{3}\)[/tex] and [tex]\(5 \sqrt{14}\)[/tex].
### Step-by-Step Solution:
1. Express the given surds:
- The first surd is [tex]\(10 \sqrt{3}\)[/tex].
- The second surd is [tex]\(5 \sqrt{14}\)[/tex].
2. Calculate [tex]\(10 \sqrt{3}\)[/tex]:
- Rewrite [tex]\(10 \sqrt{3}\)[/tex] as [tex]\(\sqrt{100} \times \sqrt{3}\)[/tex].
- [tex]\(\sqrt{100} \times \sqrt{3} = \sqrt{100 \times 3} = \sqrt{300}\)[/tex].
- Therefore, [tex]\(10 \sqrt{3} = \sqrt{300}\)[/tex]. Numerically, this is approximately 17.3205.
3. Calculate [tex]\(5 \sqrt{14}\)[/tex]:
- Rewrite [tex]\(5 \sqrt{14}\)[/tex] as [tex]\(\sqrt{25} \times \sqrt{14}\)[/tex].
- [tex]\(\sqrt{25} \times \sqrt{14} = \sqrt{25 \times 14} = \sqrt{350}\)[/tex].
- Therefore, [tex]\(5 \sqrt{14} = \sqrt{350}\)[/tex]. Numerically, this is approximately 18.7083.
4. Compare the numerical values:
- We have [tex]\(10 \sqrt{3} \approx 17.3205\)[/tex].
- We have [tex]\(5 \sqrt{14} \approx 18.7083\)[/tex].
5. Conclusion:
- Since 17.3205 is less than 18.7083, we conclude that [tex]\(10 \sqrt{3}\)[/tex] is less than [tex]\(5 \sqrt{14}\)[/tex].
So, [tex]\(10 \sqrt{3} < 5 \sqrt{14}\)[/tex]. Additionally, if we compare the squared values directly:
- [tex]\(300 < 350\)[/tex]
Thus, both the numerical evaluation and the square comparison affirm the result.
