(1 + 3) 2 + 3 | GMAT Sprint

Free Practice Question

PS · Problem Solving

705-805

Roots

$(1 + \sqrt{3})\sqrt{2 + \sqrt{3}}$

Correct answer marked below

\sqrt{2}(2 - \sqrt{3})

\sqrt{2}(2 + \sqrt{2})

\sqrt{2}(2 + \sqrt{3})

Correct

\sqrt{2}(3 + \sqrt{3})

\sqrt{3}(2 + \sqrt{3})

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Explanation

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Official Explanation

Official

Let

x = (1 + \sqrt{3})\sqrt{2 + \sqrt{3}}

.

Square both sides:

x^2 = (1 + \sqrt{3})^2 (2 + \sqrt{3})

.

Compute

(1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}

.

Then

x^2 = (4 + 2\sqrt{3})(2 + \sqrt{3})

.

Expand:

x^2 = 4(2 + \sqrt{3}) + 2\sqrt{3}(2 + \sqrt{3}) = 8 + 4\sqrt{3} + 4\sqrt{3} + 2\cdot 3 = 8 + 8\sqrt{3} + 6 = 14 + 8\sqrt{3}

.

Now factor to match a squared form:

14 + 8\sqrt{3} = 2(7 + 4\sqrt{3})

.…

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What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.