Let's simplify the expression [tex]\(\frac{36}{\sqrt{24}}\)[/tex] step by step:
1. Simplify the denominator:
[tex]\[
\sqrt{24}
\][/tex]
We can express 24 as a product of its prime factors:
[tex]\[
24 = 2^3 \cdot 3
\][/tex]
Therefore,
[tex]\[
\sqrt{24} = \sqrt{2^3 \cdot 3} = \sqrt{2^2 \cdot 2 \cdot 3} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6}
\][/tex]
2. Substitute [tex]\(\sqrt{24}\)[/tex] in the original expression:
[tex]\[
\frac{36}{\sqrt{24}} = \frac{36}{2 \sqrt{6}}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{36}{2 \sqrt{6}} = \frac{36 \div 2}{2 \sqrt{6} \div 2} = \frac{18}{\sqrt{6}}
\][/tex]
4. Rationalize the denominator:
To rationalize the denominator, multiply the numerator and the denominator by [tex]\(\sqrt{6}\)[/tex]:
[tex]\[
\frac{18}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{18 \sqrt{6}}{6}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{18 \sqrt{6}}{6} = 3 \sqrt{6}
\][/tex]
So, the expression [tex]\(\frac{36}{\sqrt{24}}\)[/tex] simplifies to [tex]\(3 \sqrt{6}\)[/tex].
Thus, the correct answer is:
(D) [tex]\(3 \sqrt{6}\)[/tex]
