What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.

12√24/3√300

1 answer

To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we will first simplify each component in the numerator and the denominator.

Starting with \(\sqrt{24}\):

\[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \]

Now substituting this back into the expression:

\[ 12\sqrt{24} = 12 \times 2\sqrt{6} = 24\sqrt{6} \]

Next, we simplify \(\sqrt{300}\):

\[ \sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3} \]

Now substituing this into the expression:

\[ 3\sqrt{300} = 3 \times 10\sqrt{3} = 30\sqrt{3} \]

Now we can rewrite the original expression:

\[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{24\sqrt{6}}{30\sqrt{3}} \]

Next, we can simplify the coefficients. The fraction \(\frac{24}{30}\) can be simplified by dividing both the numerator and denominator by 6:

\[ \frac{24}{30} = \frac{4}{5} \]

Now putting it all together, we have:

\[ \frac{24\sqrt{6}}{30\sqrt{3}} = \frac{4}{5} \cdot \frac{\sqrt{6}}{\sqrt{3}} \]

Next, we simplify the radical:

\[ \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{\frac{6}{3}} = \sqrt{2} \]

Thus, we can rewrite the expression as:

\[ \frac{4}{5} \cdot \sqrt{2} = \frac{4\sqrt{2}}{5} \]

Therefore, the quotient in simplest radical form is:

\[ \boxed{\frac{4\sqrt{2}}{5}} \]