Answer :
To determine which choice is equivalent to [tex]\(\frac{\sqrt{12}}{2 \sqrt{2}}\)[/tex], we need to simplify the expression step-by-step.
1. Simplify [tex]\(\frac{\sqrt{12}}{2 \sqrt{2}}\)[/tex]
First, recall that [tex]\(\sqrt{12}\)[/tex] can be expressed as:
[tex]\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \][/tex]
Substitute this into the original expression:
[tex]\[ \frac{2\sqrt{3}}{2 \sqrt{2}} \][/tex]
2. Cancel out common factors in the numerator and the denominator:
[tex]\[ \frac{2\sqrt{3}}{2 \sqrt{2}} = \frac{\sqrt{3}}{\sqrt{2}} \][/tex]
3. Rationalize the denominator:
Multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{3} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{6}}{2} \][/tex]
The simplified expression is:
[tex]\[ \frac{\sqrt{6}}{2} \][/tex]
4. Compare with the given choices:
- Choice A: [tex]\(\frac{\sqrt{6}}{2}\)[/tex]
- Choice B: [tex]\(\sqrt{3}\)[/tex]
- Choice C: [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]
- Choice D: [tex]\(\frac{\sqrt{6}}{4}\)[/tex]
Based on our simplification, the expression [tex]\(\frac{\sqrt{6}}{2}\)[/tex] matches Choice A.
Therefore, the equivalent expression to [tex]\(\frac{\sqrt{12}}{2 \sqrt{2}}\)[/tex] is given by:
[tex]\[ \boxed{\frac{\sqrt{6}}{2}} \][/tex]
The correct choice is A.
1. Simplify [tex]\(\frac{\sqrt{12}}{2 \sqrt{2}}\)[/tex]
First, recall that [tex]\(\sqrt{12}\)[/tex] can be expressed as:
[tex]\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \][/tex]
Substitute this into the original expression:
[tex]\[ \frac{2\sqrt{3}}{2 \sqrt{2}} \][/tex]
2. Cancel out common factors in the numerator and the denominator:
[tex]\[ \frac{2\sqrt{3}}{2 \sqrt{2}} = \frac{\sqrt{3}}{\sqrt{2}} \][/tex]
3. Rationalize the denominator:
Multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{3} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{6}}{2} \][/tex]
The simplified expression is:
[tex]\[ \frac{\sqrt{6}}{2} \][/tex]
4. Compare with the given choices:
- Choice A: [tex]\(\frac{\sqrt{6}}{2}\)[/tex]
- Choice B: [tex]\(\sqrt{3}\)[/tex]
- Choice C: [tex]\(\frac{\sqrt{6}}{\sqrt{2}}\)[/tex]
- Choice D: [tex]\(\frac{\sqrt{6}}{4}\)[/tex]
Based on our simplification, the expression [tex]\(\frac{\sqrt{6}}{2}\)[/tex] matches Choice A.
Therefore, the equivalent expression to [tex]\(\frac{\sqrt{12}}{2 \sqrt{2}}\)[/tex] is given by:
[tex]\[ \boxed{\frac{\sqrt{6}}{2}} \][/tex]
The correct choice is A.
