Question

12 radical 24/3 radical 30 State the answer in simplest radical form radilize the denominator if necessary

Answer 1

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

1. **Simplify the coefficients:**
\[
\frac{12}{3} = 4
\]

2. **Simplify the square roots:**
For \(\sqrt{24}\):
\[
\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}
\]

And for \(\sqrt{30}\), it cannot be simplified further because it has no perfect square factors.

3. **Replace in the expression:**
Now substitute back:
\[
\frac{12 \sqrt{24}}{3 \sqrt{30}} = \frac{4 \cdot 2\sqrt{6}}{\sqrt{30}} = \frac{8\sqrt{6}}{\sqrt{30}}
\]

4. **Rationalize the denominator:**
To rationalize the denominator, multiply both the numerator and denominator by \(\sqrt{30}\):
\[
\frac{8\sqrt{6} \cdot \sqrt{30}}{\sqrt{30} \cdot \sqrt{30}} = \frac{8\sqrt{180}}{30}
\]

Simplifying \(\sqrt{180}\) gives:
\[
\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}
\]

Therefore, substitute this back into the fraction:
\[
\frac{8 \times 6 \sqrt{5}}{30} = \frac{48 \sqrt{5}}{30}
\]

5. **Reduce the fraction:**
The greatest common divisor of 48 and 30 is 6, so simplify:
\[
\frac{48 \div 6 \sqrt{5}}{30 \div 6} = \frac{8\sqrt{5}}{5}
\]

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