Question

simplify
\\(\frac{16}{\sqrt{10}}\\)
hint- always look for the most simplified form :)
\\(\bigcirc\\ 8\sqrt{2}\\)
\\(\bigcirc\\ \frac{4\sqrt{10}}{10}\\)
\\(\bigcirc\\ \frac{8\sqrt{10}}{5}\\)
\\(\bigcirc\\ \frac{2\sqrt{10}}{5}\\)

Explanation:

Step1: Rationalize the denominator

To simplify \(\frac{16}{\sqrt{10}}\), we multiply the numerator and denominator by \(\sqrt{10}\) to rationalize the denominator. So we have \(\frac{16\times\sqrt{10}}{\sqrt{10}\times\sqrt{10}}\).

Step2: Simplify the denominator

Since \(\sqrt{10}\times\sqrt{10} = 10\), the expression becomes \(\frac{16\sqrt{10}}{10}\).

Step3: Reduce the fraction

We can simplify \(\frac{16\sqrt{10}}{10}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So \(\frac{16\div2\times\sqrt{10}}{10\div2}=\frac{8\sqrt{10}}{5}\)? Wait, no, wait. Wait, 16 divided by 2 is 8, 10 divided by 2 is 5? Wait, no, 16 and 10 have a GCD of 2? Wait, 16 is 2×8, 10 is 2×5. So yes. Wait, but let's check again. Wait, \(\frac{16\sqrt{10}}{10}=\frac{8\sqrt{10}}{5}\)? Wait, but let's check the options. Wait, the third option is \(\frac{8\sqrt{10}}{5}\)? Wait, no, the third option is \(\frac{8\sqrt{10}}{5}\)? Wait, the options are: first is \(8\sqrt{2}\), second is \(\frac{4\sqrt{10}}{10}\) (which can be simplified to \(\frac{2\sqrt{10}}{5}\)), third is \(\frac{8\sqrt{10}}{5}\), fourth is \(\frac{2\sqrt{10}}{5}\). Wait, let's redo the calculation.

Wait, \(\frac{16}{\sqrt{10}}\) rationalize: multiply numerator and denominator by \(\sqrt{10}\): \(\frac{16\sqrt{10}}{10}\). Then simplify \(\frac{16\sqrt{10}}{10}\) by dividing numerator and denominator by 2: \(\frac{8\sqrt{10}}{5}\). So that's the third option.

Wait, let's check the options again. The third option is \(\frac{8\sqrt{10}}{5}\). So that's the correct one.

Answer:

C. \(\frac{8\sqrt{10}}{5}\) (assuming the third option is labeled as C, but in the given options, the third one is \(\frac{8\sqrt{10}}{5}\)) Wait, the options are:

1. \(8\sqrt{2}\)

1. \(\frac{4\sqrt{10}}{10}\)

1. \(\frac{8\sqrt{10}}{5}\)

1. \(\frac{2\sqrt{10}}{5}\)

So the correct option is the third one, which is \(\frac{8\sqrt{10}}{5}\). So the answer is the third option, which is \(\frac{8\sqrt{10}}{5}\) (assuming the third option is, for example, option C, but in the given list, the third option is \(\frac{8\sqrt{10}}{5}\)). So the correct answer is the third option: \(\frac{8\sqrt{10}}{5}\).