Example 1
Completion requirements
| Example 1 |
Compare and order the following radicals from smallest to largest, without using a calculator.
\(3\sqrt[3]{5}, \sqrt[3]{{16}}, 2\sqrt[3]{{30}}, 4, \sqrt[3]{{324}}\)
\(3\sqrt[3]{5}, \sqrt[3]{{16}}, 2\sqrt[3]{{30}}, 4, \sqrt[3]{{324}}\)
Because the question requests no calculator, a good method is to convert each number into an entire radical, and then compare the radicands.
\(3\sqrt[3]{5} = \sqrt[3]{{3^3\cdot 5}} = \sqrt[3]{{135}}\)
\(\sqrt[3]{{16}} \)
\(2\sqrt[3]{{30}} = \sqrt[3]{{2^3\cdot 30}} = \sqrt[3]{{240}} \)
\(4 = \sqrt[3]{{4^3 }} = \sqrt[3]{{64}} \)
\(\sqrt[3]{{324}} \)
\(3\sqrt[3]{5} = \sqrt[3]{{3^3\cdot 5}} = \sqrt[3]{{135}}\)
\(\sqrt[3]{{16}} \)
\(2\sqrt[3]{{30}} = \sqrt[3]{{2^3\cdot 30}} = \sqrt[3]{{240}} \)
\(4 = \sqrt[3]{{4^3 }} = \sqrt[3]{{64}} \)
\(\sqrt[3]{{324}} \)
From smallest to largest, the radicals are
\(\sqrt[3]{{16}}, \sqrt[3]{{64}}, \sqrt[3]{{135}}, \sqrt[3]{{240}}, \sqrt[3]{{324}}\)
or
\(\sqrt[3]{{16}}, 4, 3\sqrt[3]{5}, 2\sqrt[3]{30}, \sqrt[3]{{324}}\)
\(\sqrt[3]{{16}}, \sqrt[3]{{64}}, \sqrt[3]{{135}}, \sqrt[3]{{240}}, \sqrt[3]{{324}}\)
or
\(\sqrt[3]{{16}}, 4, 3\sqrt[3]{5}, 2\sqrt[3]{30}, \sqrt[3]{{324}}\)
The preferred way to answer this question uses the original radicals.