Certainly! Let's simplify [tex]\(\sqrt{75}\)[/tex] step-by-step.
1. Identify the factorization of 75:
- The number 75 can be factored into [tex]\(75 = 25 \times 3\)[/tex].
2. Use the property of square roots for multiplication:
- The property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex] allows us to separate the square root into its factors.
- Thus, [tex]\(\sqrt{75} = \sqrt{25 \times 3}\)[/tex].
3. Simplify the square root of the perfect square:
- We know [tex]\(\sqrt{25} = 5\)[/tex].
- So, [tex]\(\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}\)[/tex].
4. Combine the results:
- Simplifying further, we get [tex]\(\sqrt{75} = 5\sqrt{3}\)[/tex].
Therefore, [tex]\(\sqrt{75}\)[/tex] simplifies to [tex]\(5\sqrt{3}\)[/tex].
### Verification with Numerical Approximation
- Calculating [tex]\(\sqrt{75}\)[/tex] numerically, we get [tex]\(8.660254037844387\)[/tex].
- Simplifying [tex]\(5\sqrt{3}\)[/tex], we approximate:
- [tex]\(\sqrt{3} \approx 1.732\)[/tex].
- Thus, [tex]\(5 \times 1.732 = 8.66\)[/tex].
Both approaches verify that [tex]\(\sqrt{75}\)[/tex] simplified is indeed [tex]\(5\sqrt{3}\)[/tex] and numerically approximates to [tex]\(8.660254037844387\)[/tex].
