Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

Example 1. Which of the following expressions are meaningful? Among the meaningful radicals, which are arithmetic roots? Express the radicals that are not arithmetic roots using arithmetic roots;
(1) 183\sqrt[3]{\frac{1}{8}},
(2) −9\sqrt{-9}
(3) −83\sqrt[3]{-8}

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution: (1) is an arithmetic root;
(2) is meaningless;
(3) is meaningful, but not an arithmetic root,
−83=−83=−2. \sqrt[3]{-8}=-\sqrt[3]{8}=-2 .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.