Find the square root of \(4 + 2\sqrt{3}\).
Answer (3)
( a + b ) 2 = a 2 + 2 ⋅ a ⋅ b + b 2 4 + 2 3 = 1 2 + 2 ⋅ 1 ⋅ 3 + 3 2 ⇒ a = 1 ∧ b = 3 4 + 2 3 = ( 1 + 3 ) 2 4 + 2 3 = ( 1 + 3 ) 2 = ∣1 + 3 ∣ = 1 + 3
( a + b ) 2 = a 2 + 2 ab + b 2 ( ∗ ) 4 + 2 3 = 3 + 2 3 + 1 = ( ∗ ) ( 3 ) 2 + 2 ⋅ 3 ⋅ 1 + 1 2 = ( 3 + 1 ) 2 = ∣ 3 + 1∣ = 3 + 1
The square root of 4 + 2 3 can be simplified to 1 + 3 . This is achieved by recognizing that 4 + 2 3 is a perfect square. Therefore, we find that 4 + 2 3 = 1 + 3 .
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