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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]
### 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]
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