Question

complete the following statement. use the integers that are closest to the number in the middle.
\boxed{} < \sqrt3{65} < \boxed{}
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Explanation:

Step1: Find cubes of integers

We know that \(3^3 = 27\), \(4^3=64\), \(5^3 = 125\).

Step2: Compare with \(\sqrt[3]{65}\)

Since \(4^3=64\) and \(5^3 = 125\), and \(64<65<125\), taking cube - roots (the cube - root function \(y = \sqrt[3]{x}\) is an increasing function, so if \(a < b\), then \(\sqrt[3]{a}<\sqrt[3]{b}\)), we get \(\sqrt[3]{64}<\sqrt[3]{65}<\sqrt[3]{125}\).
Since \(\sqrt[3]{64} = 4\) and \(\sqrt[3]{125}=5\), we have \(4<\sqrt[3]{65}<5\).

Answer:

\(4<\sqrt[3]{65}<5\)