To solve the expression [tex]\(\frac{9 \sqrt{15} + 3 \sqrt{10} - 3 \sqrt{11} + \sqrt{14}}{25}\)[/tex], let's break it down step-by-step.
1. Identify and Simplify Each Term:
- [tex]\(9 \sqrt{15}\)[/tex]
- [tex]\(3 \sqrt{10}\)[/tex]
- [tex]\(-3 \sqrt{11}\)[/tex]
- [tex]\(\sqrt{14}\)[/tex]
2. Calculate the Numeric Values of Each Term:
- [tex]\(9 \sqrt{15} \approx 34.856850115866756\)[/tex]
- [tex]\(3 \sqrt{10} \approx 9.486832980505138\)[/tex]
- [tex]\(-3 \sqrt{11} \approx -9.9498743710662\)[/tex]
- [tex]\(\sqrt{14} \approx 3.7416573867739413\)[/tex]
3. Sum These Values to Find the Numerator:
Let's add these calculated values together:
[tex]\[
34.856850115866756 + 9.486832980505138 - 9.9498743710662 + 3.7416573867739413 = 38.135466112079634
\][/tex]
So, the numerator is approximately [tex]\(38.135466112079634\)[/tex].
4. Divide by the Denominator:
The original expression has a denominator of 25. So, we need to divide the sum of the terms by 25:
[tex]\[
\frac{38.135466112079634}{25} = 1.5254186444831854
\][/tex]
Hence, the result of the expression [tex]\(\frac{9 \sqrt{15} + 3 \sqrt{10} - 3 \sqrt{11} + \sqrt{14}}{25}\)[/tex] is approximately [tex]\(1.5254186444831854\)[/tex].
