Evaluate

Problem

√(,(1−√(,3)/2)/2)

Solution

1. Simplify the numerator inside the large square root by finding a common denominator.

(1−√(,3)/2)/2=(2−√(,3))/2/2

2. Divide the fractions by multiplying the denominator of the numerator by the main denominator.

(2−√(,3))/2/2=(2−√(,3))/4

3. Apply the square root to both the numerator and the denominator separately.

√(,(2−√(,3))/4)=√(,2−√(,3))/√(,4)

4. Simplify the denominator.

√(,2−√(,3))/√(,4)=√(,2−√(,3))/2

5. Multiply the numerator and denominator by √(,2) to transform the expression into a perfect square trinomial under the radical.

(√(,2)⋅√(,2−√(,3)))/(2⋅√(,2))=√(,4−2√(,3))/(2√(,2))

6. Recognize that the expression 4−2√(,3) is a perfect square, specifically (√(,3)−1)2

4−2√(,3)=3−2√(,3)+1=(√(,3)−1)2

7. Substitute the perfect square back into the radical and simplify.

√(,(√(,3)−1)2)/(2√(,2))=(√(,3)−1)/(2√(,2))

8. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

((√(,3)−1)√(,2))/(2√(,2)√(,2))=(√(,6)−√(,2))/4

Final Answer

√(,(1−√(,3)/2)/2)=(√(,6)−√(,2))/4

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