Evaluate

Problem

(3+√(,5))/(2−√(,5))

Solution

1. Identify the need to rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is 2+√(,5)

(3+√(,5))/(2−√(,5))⋅(2+√(,5))/(2+√(,5))

2. Expand the numerator using the FOIL method (First, Outer, Inner, Last).

(3)*(2)+(3)*(√(,5))+(√(,5))*(2)+(√(,5))*(√(,5))

6+3√(,5)+2√(,5)+5

11+5√(,5)

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

(2)2−(√(,5))2

4−5

−1

4. Simplify the expression by dividing the expanded numerator by the simplified denominator.

(11+5√(,5))/(−1)

−11−5√(,5)

Final Answer

(3+√(,5))/(2−√(,5))=−11−5√(,5)

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