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Which choices are equivalent to the quotient below? Check all that apply.

[tex]\[
\frac{\sqrt{16}}{\sqrt{8}}
\][/tex]

A. [tex]\(\frac{\sqrt{6}}{2}\)[/tex]

B. [tex]\(\sqrt{2}\)[/tex]

C. [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]

D. 2

E. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]

F. [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]


Sagot :

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]