Free Practice Question

PS · Problem Solving

705-805

Roots

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

Answer Choices

Correct answer marked below

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

Correct

B

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

C

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

D

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

E

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

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Explanation

Preview

Official Explanation

Official

Let

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

, so

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

.

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

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

…

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What this question tests

What it tests

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

Common trap

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

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(1+3)2+3=?(1+\sqrt{3}) \sqrt{2+\sqrt{3}} = ?(1+3​)2+3​​=?