1. Transform:

[tex]\[ \sqrt{14+\sqrt{180}} \][/tex]

a) \(\sqrt{5} + 3\)

b) \(\sqrt{5} - 3\)

c) \(3 - \sqrt{5}\)

d) \(\sqrt{3} + 5\)

e) [tex]\(\sqrt{3} - 5\)[/tex]



Answer :

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]