Evaluate

Problem

(2√(,3)+√(,6))/(4√(,3)−√(,6))

Solution

1. Identify the conjugate of the denominator to rationalize the expression. The conjugate of 4√(,3)−√(,6) is 4√(,3)+√(,6)

2. Multiply both the numerator and the denominator by this conjugate.

((2√(,3)+√(,6))*(4√(,3)+√(,6)))/((4√(,3)−√(,6))*(4√(,3)+√(,6)))

3. Expand the numerator using the FOIL method.

(2√(,3))*(4√(,3))+(2√(,3))*(√(,6))+(√(,6))*(4√(,3))+(√(,6))*(√(,6))

8*(3)+2√(,18)+4√(,18)+6

24+6√(,18)+6

30+6√(,9⋅2)

30+18√(,2)

4. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2

(4√(,3))2−(√(,6))2

16*(3)−6

48−6

42

5. Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 6

(30+18√(,2))/42

(6*(5+3√(,2)))/(6*(7))

(5+3√(,2))/7

Final Answer

(2√(,3)+√(,6))/(4√(,3)−√(,6))=(5+3√(,2))/7

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