Simplify (1 + 2√3)/(2 – √3)

Explanation

Step 1: Identify conjugate of denominator
Denominator: 2 – √3
Conjugate: 2 + √3

Step 2: Multiply numerator and denominator by conjugate
= [(1 + 2√3)(2 + √3)]/[(2 – √3)(2 + √3)]

Step 3: Expand numerator (FOIL)
(1 + 2√3)(2 + √3)
= 1(2) + 1(√3) + 2√3(2) + 2√3(√3)
= 2 + √3 + 4√3 + 2(3)
= 2 + √3 + 4√3 + 6
= 8 + 5√3

Step 4: Expand denominator (difference of squares)
(2 – √3)(2 + √3) = 2² – (√3)²
= 4 – 3
= 1

Step 5: Final answer
(8 + 5√3)/1 = 8 + 5√3