Evaluate

Problem

√(,14+6√(,5))+3+√(,5)

Solution

1. Identify the nested radical √(,14+6√(,5)) and attempt to simplify it by expressing the radicand as a perfect square of the form (a+b)^2=a^2+b^2+2*a*b

2. Rewrite the term 6√(,5) as 2*(3√(,5)) to find the 2*a*b component, where a=3 and b=√(,5)

3. Verify the sum of the squares a^2+b^2 by calculating 3^2+(√(,5))^2=9+5=14 which matches the constant term in the radicand.

4. Simplify the nested radical by substituting the perfect square form:

√(,14+6√(,5))=√(,(3+√(,5))^2)

5. Extract the square root:

√(,(3+√(,5))^2)=3+√(,5)

6. Substitute this simplified expression back into the original problem:

(3+√(,5))+3+√(,5)

7. Combine like terms by adding the integers and the radicals separately:

3+3=6

√(,5)+√(,5)=2√(,5)

Final Answer

√(,14+6√(,5))+3+√(,5)=6+2√(,5)

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