Answer :
To determine which choices are equivalent to the quotient [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex], let us simplify the given quotient step by step.
First, consider the quotient:
[tex]\[ \frac{\sqrt{12}}{\sqrt{6}} \][/tex]
We can combine the square roots:
[tex]\[ \frac{\sqrt{12}}{\sqrt{6}} = \sqrt{\frac{12}{6}} \][/tex]
Now, simplify the fraction inside the square root:
[tex]\[ \frac{12}{6} = 2 \][/tex]
So, we have:
[tex]\[ \sqrt{\frac{12}{6}} = \sqrt{2} \][/tex]
Thus, the simplified form of the quotient is:
[tex]\[ \sqrt{2} \][/tex]
Next, we will compare this result with each of the given choices:
Choice A: [tex]\(\frac{\sqrt{6}}{2}\)[/tex]
Given as [tex]\(\frac{\sqrt{6}}{2}\)[/tex], we need to determine if this is equal to [tex]\(\sqrt{2}\)[/tex]:
Simplify [tex]\(\frac{\sqrt{6}}{2}\)[/tex]:
[tex]\[ \frac{\sqrt{6}}{2} \neq \sqrt{2} \][/tex]
So, Choice A is not correct.
Choice B: [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]
Simplify [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]:
[tex]\[ \frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3} \][/tex]
So, [tex]\(\sqrt{3} \neq \sqrt{2}\)[/tex]:
Choice B is not correct.
Choice C: [tex]\(\sqrt{2}\)[/tex]
Given that [tex]\(\sqrt{2}\)[/tex] is already simplified, it is clear:
[tex]\[ \sqrt{2} = \sqrt{2} \][/tex]
So, Choice C is correct.
Choice D: [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]
Simplify [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]:
[tex]\[ \frac{\sqrt{4}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \][/tex]
So, [tex]\(\sqrt{2} = \sqrt{2}\)[/tex]:
Choice D is correct.
Choice E: 2
Given the constant 2:
[tex]\[ 2 \neq \sqrt{2} \][/tex]
So, Choice E is not correct.
Choice F: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
Simplify [tex]\(\frac{2}{\sqrt{3}}\)[/tex]:
[tex]\[ \frac{2}{\sqrt{3}} \neq \sqrt{2} \][/tex]
So, Choice F is not correct.
Based on the simplifications, the correct choices that are equivalent to [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex] are:
[tex]\[ \boxed{C} \quad \boxed{D} \][/tex]
Thus, choices C and D are equivalent to the quotient [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex].
First, consider the quotient:
[tex]\[ \frac{\sqrt{12}}{\sqrt{6}} \][/tex]
We can combine the square roots:
[tex]\[ \frac{\sqrt{12}}{\sqrt{6}} = \sqrt{\frac{12}{6}} \][/tex]
Now, simplify the fraction inside the square root:
[tex]\[ \frac{12}{6} = 2 \][/tex]
So, we have:
[tex]\[ \sqrt{\frac{12}{6}} = \sqrt{2} \][/tex]
Thus, the simplified form of the quotient is:
[tex]\[ \sqrt{2} \][/tex]
Next, we will compare this result with each of the given choices:
Choice A: [tex]\(\frac{\sqrt{6}}{2}\)[/tex]
Given as [tex]\(\frac{\sqrt{6}}{2}\)[/tex], we need to determine if this is equal to [tex]\(\sqrt{2}\)[/tex]:
Simplify [tex]\(\frac{\sqrt{6}}{2}\)[/tex]:
[tex]\[ \frac{\sqrt{6}}{2} \neq \sqrt{2} \][/tex]
So, Choice A is not correct.
Choice B: [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]
Simplify [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]:
[tex]\[ \frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3} \][/tex]
So, [tex]\(\sqrt{3} \neq \sqrt{2}\)[/tex]:
Choice B is not correct.
Choice C: [tex]\(\sqrt{2}\)[/tex]
Given that [tex]\(\sqrt{2}\)[/tex] is already simplified, it is clear:
[tex]\[ \sqrt{2} = \sqrt{2} \][/tex]
So, Choice C is correct.
Choice D: [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]
Simplify [tex]\(\frac{\sqrt{4}}{\sqrt{2}}\)[/tex]:
[tex]\[ \frac{\sqrt{4}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \][/tex]
So, [tex]\(\sqrt{2} = \sqrt{2}\)[/tex]:
Choice D is correct.
Choice E: 2
Given the constant 2:
[tex]\[ 2 \neq \sqrt{2} \][/tex]
So, Choice E is not correct.
Choice F: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
Simplify [tex]\(\frac{2}{\sqrt{3}}\)[/tex]:
[tex]\[ \frac{2}{\sqrt{3}} \neq \sqrt{2} \][/tex]
So, Choice F is not correct.
Based on the simplifications, the correct choices that are equivalent to [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex] are:
[tex]\[ \boxed{C} \quad \boxed{D} \][/tex]
Thus, choices C and D are equivalent to the quotient [tex]\(\frac{\sqrt{12}}{\sqrt{6}}\)[/tex].
