Certainly! To simplify and express the ratio [tex]\(\sqrt{8}: \sqrt{200}: \sqrt{72}\)[/tex] in its simplest form, we follow a detailed process. Here is a step-by-step solution:
1. Calculate the Square Roots:
- [tex]\(\sqrt{8}\)[/tex]
- [tex]\(\sqrt{200}\)[/tex]
- [tex]\(\sqrt{72}\)[/tex]
2. Values of the Square Roots:
- [tex]\(\sqrt{8} \approx 2.8284271247461903\)[/tex]
- [tex]\(\sqrt{200} \approx 14.142135623730951\)[/tex]
- [tex]\(\sqrt{72} \approx 8.48528137423857\)[/tex]
3. Establish the Ratio:
- The initial ratio is approximately [tex]\(2.8284271247461903 : 14.142135623730951 : 8.48528137423857\)[/tex]
4. Find the Greatest Common Divisors (GCD) to simplify the terms:
- Since [tex]\( \sqrt{8} \)[/tex] and [tex]\( \sqrt{200} \)[/tex] roughly share a factor in common, divide each term by this common factor.
5. Simplify the Terms:
- The term simplifications are as follows:
- [tex]\( 2.8284271247461903 / 2.8284271247461903 = 1 \)[/tex]
- [tex]\( 14.142135623730951 / 2.8284271247461903 \approx 5 \)[/tex]
- [tex]\( 8.48528137423857 / 2.8284271247461903 \approx 3 \)[/tex]
6. Final Simplified Ratio:
- Combining these results, we get:
- [tex]\(1 : 5 : 3\)[/tex]
Thus, the simplified form of the ratio [tex]\( \sqrt{8}: \sqrt{200}: \sqrt{72} \)[/tex] is [tex]\(1 : 5 : 3\)[/tex].
