Question

which integer is closest to \\(\sqrt3{23}\\)?

Explanation:

Step1: Recall cube values

We know the cubes of integers: \(2^3 = 8\), \(3^3=27\).

Step2: Compare distances

The value of \(\sqrt[3]{23}\) is between \(2\) and \(3\) because \(2^3 = 8<23<27 = 3^3\). Now, calculate the distance from \(23\) to \(8\) and to \(27\): \(23 - 8=15\), \(27 - 23 = 4\). Since \(4<15\), \(23\) is closer to \(27\) (which is \(3^3\)) than to \(8\) (which is \(2^3\)). So \(\sqrt[3]{23}\) is closer to \(3\) than to \(2\).

Answer:

\(3\)