Sure, let's analyze each given option step-by-step and see which are equivalent to the given quotient [tex]\(\frac{\sqrt{16}}{\sqrt{8}}\)[/tex].
First, let's simplify the given quotient:
[tex]\[
\frac{\sqrt{16}}{\sqrt{8}}
\][/tex]
Simplify the square roots:
[tex]\[
\sqrt{16} = 4
\][/tex]
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}
\][/tex]
Now substitute these back into the quotient:
[tex]\[
\frac{4}{2\sqrt{2}}
\][/tex]
Simplify further:
[tex]\[
\frac{4}{2\sqrt{2}} = \frac{4}{2} \cdot \frac{1}{\sqrt{2}} = 2 \cdot \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}
\][/tex]
Therefore, [tex]\(\frac{\sqrt{16}}{\sqrt{8}} = \sqrt{2}\)[/tex].
Now we'll compare each option to [tex]\(\sqrt{2}\)[/tex]:
Option A: [tex]\(\frac{\sqrt{6}}{2}\)[/tex]
This does not simplify to [tex]\(\sqrt{2}\)[/tex].
Option B: [tex]\(\sqrt{2}\)[/tex]
This is equivalent to [tex]\(\sqrt{2}\)[/tex].
Option C: [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]
Simplify the fraction:
[tex]\[
\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3}
\][/tex]
This does not simplify to [tex]\(\sqrt{2}\)[/tex].
Option D: 2
This is not equivalent to [tex]\(\sqrt{2}\)[/tex].
Option E: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
This does not simplify to [tex]\(\sqrt{2}\)[/tex].
Option F: [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]
Simplify the fraction:
[tex]\[
\frac{\sqrt{4}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = 2 \cdot \frac{1}{\sqrt{2}} = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2}
\][/tex]
This is equivalent to [tex]\(\sqrt{2}\)[/tex].
Thus, the choices equivalent to the given quotient [tex]\(\frac{\sqrt{16}}{\sqrt{8}}\)[/tex] are:
[tex]\[
\boxed{B, F}
\][/tex]
