To solve the given transformation of the expression \(\sqrt{14 + \sqrt{180}}\), we will systematically compare it with each of the provided options to identify which one it matches.
Given expression:
[tex]\[
\sqrt{14 + \sqrt{180}}
\][/tex]
Step-by-Step Transformation:
1. Calculate the value of \(\sqrt{14 + \sqrt{180}}\):
We need to compute the value of \(\sqrt{14 + \sqrt{180}}\). The numerical result is approximately 5.236.
2. Compare with the given options:
- Option a: \(\sqrt{5} + 3\)
[tex]\[
\sqrt{5} \approx 2.236
\][/tex]
So,
[tex]\[
\sqrt{5} + 3 \approx 2.236 + 3 = 5.236
\][/tex]
This value is approximately 5.236, which is the same as the computed value of \(\sqrt{14 + \sqrt{180}}\).
- Option b: \(\sqrt{5} - 3\)
[tex]\[
\sqrt{5} - 3 \approx 2.236 - 3 = -0.764
\][/tex]
This value is approximately -0.764, which does not match the value of \(\sqrt{14 + \sqrt{180}}\).
- Option c: \(3 - \sqrt{5}\)
[tex]\[
3 - \sqrt{5} \approx 3 - 2.236 = 0.764
\][/tex]
This value is approximately 0.764, which does not match the value of \(\sqrt{14 + \sqrt{180}}\).
- Option d: \(\sqrt{3} + 5\)
[tex]\[
\sqrt{3} \approx 1.732
\][/tex]
So,
[tex]\[
\sqrt{3} + 5 \approx 1.732 + 5 = 6.732
\][/tex]
This value is approximately 6.732, which does not match the value of \(\sqrt{14 + \sqrt{180}}\).
- Option e: \(\sqrt{3} - 5\)
[tex]\[
\sqrt{3} - 5 \approx 1.732 - 5 = -3.268
\][/tex]
This value is approximately -3.268, which does not match the value of \(\sqrt{14 + \sqrt{180}}\).
Conclusion:
After comparing all the given options, the value that matches the transformation of \(\sqrt{14 + \sqrt{180}}\) is \(\sqrt{5} + 3\).
So, the correct transformation is:
[tex]\[
\boxed{\sqrt{5} + 3}
\][/tex]
