Question

what is the following quotient? \\(\frac{\sqrt{96}}{\sqrt{8}}\\) options: 12, 4, \\(2\sqrt{22}\\), \\(2\sqrt{3}\\)

Explanation:

Step1: Simplify the square roots

We know that \(\sqrt{96}=\sqrt{16\times6}\) and \(\sqrt{8}=\sqrt{4\times2}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{96}=\sqrt{16}\times\sqrt{6} = 4\sqrt{6}\) and \(\sqrt{8}=\sqrt{4}\times\sqrt{2}=2\sqrt{2}\). But maybe a better way is to use the property \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) (\(a\geq0,b > 0\)). So \(\frac{\sqrt{96}}{\sqrt{8}}=\sqrt{\frac{96}{8}}\).

Step2: Calculate the fraction inside the square root

Calculate \(\frac{96}{8}=12\)? Wait, no, \(96\div8 = 12\)? Wait, no, \(8\times12 = 96\)? Wait, no, \(8\times12=96\), but wait, let's check again. Wait, \(96\div8 = 12\)? Wait, no, \(8\times12 = 96\), but then \(\sqrt{12}\) can be simplified. Wait, \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\). Wait, let's do it step by step. \(\frac{\sqrt{96}}{\sqrt{8}}=\sqrt{\frac{96}{8}}=\sqrt{12}\). Then simplify \(\sqrt{12}\): \(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}\).

Answer:

\(2\sqrt{3}\) (the option with \(2\sqrt{3}\))