To solve the expression [tex]\(-2 \sqrt{3} + \sqrt{75}\)[/tex], let's go through the steps to simplify each term and combine them:
1. Simplify [tex]\(\sqrt{75}\)[/tex]:
- Start by expressing 75 as the product of its prime factors: [tex]\(75 = 25 \times 3\)[/tex].
- We know that [tex]\( \sqrt{25 \times 3} \)[/tex] can be simplified to [tex]\(\sqrt{25} \times \sqrt{3}\)[/tex].
- Since [tex]\( \sqrt{25} = 5 \)[/tex], we have [tex]\(\sqrt{75} = 5\sqrt{3}\)[/tex].
2. Combine the terms [tex]\(-2 \sqrt{3} + 5 \sqrt{3}\)[/tex]:
- We can factor out [tex]\( \sqrt{3} \)[/tex] as it is a common term.
- This leaves us with [tex]\(-2 \sqrt{3} + 5 \sqrt{3} = (5 - 2) \sqrt{3} = 3 \sqrt{3}\)[/tex].
3. Summary of the intermediate and final results:
- The term [tex]\(-2 \sqrt{3}\)[/tex] evaluates numerically to approximately [tex]\(-3.464\)[/tex].
- The term [tex]\(5 \sqrt{3}\)[/tex] evaluates numerically to approximately [tex]\(8.660\)[/tex].
- Combining these terms yields the final result: [tex]\(3 \sqrt{3}\)[/tex], which evaluates numerically to approximately [tex]\(5.196\)[/tex].
Therefore, the step-by-step solution confirms that [tex]\(-2 \sqrt{3} + 5 \sqrt{3} = 3 \sqrt{3}\)[/tex].
