Answer :
To determine which choice is equivalent to the fraction [tex]\(\frac{6}{\sqrt{2}}\)[/tex], we need to rationalize the denominator and simplify the expression. Here is the step-by-step process:
1. Write the original fraction:
[tex]\[ \frac{6}{\sqrt{2}} \][/tex]
2. Rationalize the denominator:
To rationalize the denominator, multiply both the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6 \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \][/tex]
3. Simplify the denominator:
Since [tex]\(\sqrt{2} \cdot \sqrt{2} = 2\)[/tex], the fraction becomes:
[tex]\[ \frac{6 \sqrt{2}}{2} \][/tex]
4. Simplify the fraction:
Divide the numerator by the denominator:
[tex]\[ \frac{6 \sqrt{2}}{2} = 3 \sqrt{2} \][/tex]
So, [tex]\(\frac{6}{\sqrt{2}}\)[/tex] simplifies to [tex]\(3 \sqrt{2}\)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{D. \ 3 \sqrt{2}} \][/tex]
1. Write the original fraction:
[tex]\[ \frac{6}{\sqrt{2}} \][/tex]
2. Rationalize the denominator:
To rationalize the denominator, multiply both the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6 \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \][/tex]
3. Simplify the denominator:
Since [tex]\(\sqrt{2} \cdot \sqrt{2} = 2\)[/tex], the fraction becomes:
[tex]\[ \frac{6 \sqrt{2}}{2} \][/tex]
4. Simplify the fraction:
Divide the numerator by the denominator:
[tex]\[ \frac{6 \sqrt{2}}{2} = 3 \sqrt{2} \][/tex]
So, [tex]\(\frac{6}{\sqrt{2}}\)[/tex] simplifies to [tex]\(3 \sqrt{2}\)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{D. \ 3 \sqrt{2}} \][/tex]
