PS arithmetic

kyuhunl wrote:$$\left(1+\sqrt{3}\right)\sqrt{2+\sqrt{3}}=?$$

A. $$\sqrt{2}\left(2-\sqrt{3}\right)$$
B. $$\sqrt{2}\left(2+\sqrt{2}\right)$$
C. $$\sqrt{2}\left(2+\sqrt{3}\right)$$
D. $$\sqrt{2}\left(3+\sqrt{3}\right)$$
E. $$\sqrt{3}\left(2+\sqrt{3}\right)$$

OA: C

Recall that if x > 0, then x = √(x^2).

Since 1 + √3 > 0, 1 + √3 = √[( 1+ √3)^2] = √[1 + 2√3 + 3] = √(4 + 2√3).

Therefore,

(1 + √3)√(2 + √3)

√(4 + 2√3)√(2 + √3)

√[(4 + 2√3)(2 + √3)]

√[2(2 + √3)(2 + √3)]

√[2(2 + √3)^2]

√2(2 + √3)

Alternate Solution:

Let’s let (1 + √3)√[2 + √3] = x and square each side:

(1 + √3)√[2 + √3] = x

[(1 + √3)^2](2 + √3) = x^2

(1 + 2√3 + 3)(2 + √3) = x^2

(4 + 2√3)(2 + √3) = x^2

2(2 + √3)(2 + √3) = x^2

2(2 + √3)^2 = x^2

Taking the square root of each side, we conclude that x = ±√2(2 + √3). Going back to the original expression that was given to us, we observe that both factors of (1 + √3)√[2 + √3] are positive; therefore, we can eliminate the negative root and conclude that x = √2(2 + √3).

Answer: C