12 radical 24/3 radical 30 state to answer and simplest radical form

Answer 1

To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{30}}\), we can start by simplifying both the numerator and denominator:

1. Numerator: \[ 12\sqrt{24} = 12\sqrt{4 \cdot 6} = 12 \cdot 2\sqrt{6} = 24\sqrt{6} \]

2. Denominator: \[ 3\sqrt{30} \text{ is already in simplest form.} \]

Now we substitute these simplified forms back into the expression: \[ \frac{12\sqrt{24}}{3\sqrt{30}} = \frac{24\sqrt{6}}{3\sqrt{30}} \]

Next, simplify the fraction: \[ \frac{24}{3} = 8 \] Thus, we get: \[ \frac{12\sqrt{24}}{3\sqrt{30}} = \frac{24\sqrt{6}}{3\sqrt{30}} = 8 \cdot \frac{\sqrt{6}}{\sqrt{30}} \]

Now, simplify \(\frac{\sqrt{6}}{\sqrt{30}}\): \[ \frac{\sqrt{6}}{\sqrt{30}} = \sqrt{\frac{6}{30}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \]

Therefore, our expression becomes: \[ 8 \cdot \frac{1}{\sqrt{5}} = \frac{8}{\sqrt{5}} \]

To express this in simplest radical form, we rationalize the denominator: \[ \frac{8}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{8\sqrt{5}}{5} \]

Thus, the final answer in simplest radical form is: \[ \frac{8\sqrt{5}}{5} \]