$$\left(1+\sqrt{3}\right)\sqrt{2+\sqrt{3}}=?$$
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
PS arithmetic
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- fskilnik@GMATH
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Beautiful problem, kyuhunl. Congrats!kyuhunl wrote:$$\left(1+\sqrt{3}\right)\sqrt{2+\sqrt{3}}=?$$
$$A. \sqrt{2}\left(2-\sqrt{3}\right)\,\,\,\,\,\,\,B. \sqrt{2}\left(2+\sqrt{2}\right)\,\,\,\,\,\,\,C. \sqrt{2}\left(2+\sqrt{3}\right)\,\,\,\,\,\,\,D. \sqrt{2}\left(3+\sqrt{3}\right)\,\,\,\,\,\,\,E. \sqrt{3}\left(2+\sqrt{3}\right)$$
$$? = \left( {1 + \sqrt 3 } \right)\sqrt {2 + \sqrt 3 } = x\,\,\,\,\,\left( {x > 0} \right)$$
$${x^2} = {\left( {1 + \sqrt 3 } \right)^2}\left( {2 + \sqrt 3 } \right) = \left( {1 + 2\sqrt 3 + 3} \right)\left( {2 + \sqrt 3 } \right)$$
$${x^2} = 2\left( {2 + \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)\,\,\,\,\,\mathop \Rightarrow \limits^{x\,\, > \,\,0} \,\,\,\,\,? = \sqrt 2 \left( {2 + \sqrt 3 } \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{C}} \right)$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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An alternate approach is to BALLPARK.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. $$\sqrt{3}\left(2+\sqrt{3}\right)$$
√2 ≈ 1.4
√3 ≈ 1.7
(1+√3) * √(2 + √3) ≈ (2.7)(√3.7) = (2.7)(a bit less than 2) = a bit less than 5.4.
When we evaluate the answer choices, only C is a bit less than 5.4:
√2(2 + √3) ≈ (1.4)(3.7) = 5.18.
The correct answer is C.
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Recall that if x > 0, then x = √(x^2).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
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
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