Question

complete the following statement. use the integers that are closer middle.
\boxed{} < \sqrt3{21} < \boxed{}
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Explanation:

Step1: Find cubes of integers

We know that \(2^3 = 8\) and \(3^3=27\).

Step2: Compare with \(\sqrt[3]{21}\)

Since \(8<21 < 27\), taking cube - root on all parts of the inequality (the cube - root function \(y = \sqrt[3]{x}\) is an increasing function, so the direction of the inequality remains the same when we take the cube - root), we get \(\sqrt[3]{8}<\sqrt[3]{21}<\sqrt[3]{27}\). And since \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{27}=3\), we have \(2<\sqrt[3]{21}<3\).

Answer:

\(2<\sqrt[3]{21}<3\)