To solve the problem of determining which choices are equivalent to the given quotient [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex], follow these steps:
### Step 1: Simplify the Quotient
First, let's simplify the quotient [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex]:
[tex]\[
\frac{\sqrt{12}}{\sqrt{6}} = \sqrt{\frac{12}{6}} = \sqrt{2}
\][/tex]
Thus, the simplified form of the quotient is [tex]\(\sqrt{2}\)[/tex].
### Step 2: Evaluate Each Choice
Next, we need to evaluate each choice and determine if it is equivalent to [tex]\(\sqrt{2}\)[/tex]:
- Choice A: [tex]\(\sqrt{2}\)[/tex]
[tex]\[
\text{This is directly } \sqrt{2}, \text{ which matches the simplified form of the quotient. Thus, Choice A is correct.}
\][/tex]
- Choice B: [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]
[tex]\[
\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3}
\][/tex]
[tex]\[
\text{This is } \sqrt{3}, \text{ which is not equivalent to } \sqrt{2}. \text{ Thus, Choice B is incorrect.}
\][/tex]
- Choice C: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
[tex]\[
\frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\][/tex]
[tex]\[
\text{This does not simplify to } \sqrt{2}. \text{ Thus, Choice C is incorrect.}
\][/tex]
- Choice D: [tex]\(\frac{\sqrt{6}}{2}\)[/tex]
[tex]\[
\frac{\sqrt{6}}{2}
\][/tex]
[tex]\[
\text{This neither simplifies to } \sqrt{2} \text{. Thus, Choice D is incorrect.}
\][/tex]
- Choice E: [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]
[tex]\[
\frac{\sqrt{4}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\][/tex]
[tex]\[
\text{This equals } \sqrt{2}, \text{ which matches the simplified form of the quotient. Thus, Choice E is correct.}
\][/tex]
- Choice F: 2
[tex]\[
2
\][/tex]
[tex]\[
2 \text{ is not equivalent to } \sqrt{2}. \text{ Thus, Choice F is incorrect.}
\][/tex]
### Final Result
After evaluating all choices, the ones that are equivalent to the quotient [tex]\(\frac{\sqrt{12}}{\sqrt{6}} = \sqrt{2}\)[/tex] are:
- Choice A: [tex]\(\sqrt{2}\)[/tex]
- Choice E: [tex]\(\frac{\sqrt{4}}{\sqrt{2}} = \sqrt{2}\)[/tex]
Thus, the correct choices are:
[tex]\[
\boxed{A \text{ and } E}
\][/tex]
