Question

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

(12sqrt(24))/(3sqrt(300))

Answer 1

thats wrong

Answer 2

To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we start by reducing the coefficients and the square roots.

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

2. **Simplify the square roots**:

- For \(\sqrt{24}\):
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
\]

- For \(\sqrt{300}\):
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3}
\]

3. **Substituting back into the expression**:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{4 \cdot 2\sqrt{6}}{10\sqrt{3}} = \frac{8\sqrt{6}}{10\sqrt{3}}
\]

4. **Simplify the fraction**:
\[
\frac{8}{10} = \frac{4}{5}
\]
Thus,
\[
\frac{8\sqrt{6}}{10\sqrt{3}} = \frac{4\sqrt{6}}{5\sqrt{3}}
\]

5. **Rationalize the denominator**:
To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\):
\[
\frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{18}}{5 \cdot 3} = \frac{4\sqrt{18}}{15}
\]

6. **Simplify the square root in the numerator**:
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
Therefore, substituting back gives:
\[
\frac{4 \cdot 3\sqrt{2}}{15} = \frac{12\sqrt{2}}{15}
\]

7. **Final simplification**:
\[
\frac{12}{15} = \frac{4}{5}
\]
Thus, final expression is:
\[
\frac{4\sqrt{2}}{5}
\]

The final answer in simplest radical form is:
\[
\frac{4\sqrt{2}}{5}
\]

Answer 3

I apologize for any confusion. Let's work through the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\) step-by-step to find the correct quotient:

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

2. **Simplifying the square roots**:

- For \(\sqrt{24}\):
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
\]

- For \(\sqrt{300}\):
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3}
\]

3. **Substituting back into the expression**:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{4 \cdot 2\sqrt{6}}{10\sqrt{3}} = \frac{8\sqrt{6}}{10\sqrt{3}}
\]

4. **Simplifying the fraction**:
\[
\frac{8}{10} = \frac{4}{5}
\]
Thus,
\[
\frac{8\sqrt{6}}{10\sqrt{3}} = \frac{4\sqrt{6}}{5\sqrt{3}}
\]

5. **Rationalizing the denominator**:
To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\):
\[
\frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{18}}{15}
\]

6. **Simplifying the square root**:
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
So, we have:
\[
\frac{4 \cdot 3\sqrt{2}}{15} = \frac{12\sqrt{2}}{15}
\]

7. **Final simplification**:
Simplifying \(\frac{12}{15}\) gives:
\[
\frac{12}{15} = \frac{4}{5}
\]
Thus, the final expression is:
\[
\frac{4\sqrt{2}}{5}
\]

Therefore, the correctly simplified expression is:
\[
\frac{4\sqrt{2}}{5}
\]

If there was another error or confusion in your interpretation, please clarify and I'll correct that!