Question

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

12√24/3√300

Answer 1

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:

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