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138(A) Является ли число sqrt(52-14sqrt3)+sqrt3 рациональным?

138(A) 138(A) \newlineЯвляется ли число 52−143+3 \sqrt{52-14 \sqrt{3}}+\sqrt{3} рациональным?

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Q. 138(A) 138(A) \newlineЯвляется ли число 52−143+3 \sqrt{52-14 \sqrt{3}}+\sqrt{3} рациональным?
  1. Simplify expression inside square root: First, let's simplify the expression inside the square root: 52−143\sqrt{52 - 14\sqrt{3}}.
  2. Express 5252 as product: 5252 can be written as 4×134 \times 13, and since 44 is a perfect square, we can take the square root of 44 out of the square root.\newlineSo, 52−143=4×13−143=4×13−14×3\sqrt{52 - 14\sqrt{3}} = \sqrt{4 \times 13 - 14\sqrt{3}} = \sqrt{4} \times \sqrt{13} - \sqrt{14} \times \sqrt{\sqrt{3}}.

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