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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 point)

5 answers

To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we can start by simplifying the square roots and then simplifying the fraction.

  1. Simplify the square roots:

    \(\sqrt{24}\) can be simplified as follows:

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

    Therefore, we replace \(\sqrt{24}\) in the expression:

    \[ 12\sqrt{24} = 12(2\sqrt{6}) = 24\sqrt{6} \]

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

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

    Thus we replace \(\sqrt{300}\):

    \[ 3\sqrt{300} = 3(10\sqrt{3}) = 30\sqrt{3} \]

  2. Substitute back into the expression:

    Now we have:

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

  3. Simplify the fraction:

    We can simplify the coefficients \(24\) and \(30\) by finding the greatest common divisor (GCD), which is \(6\):

    \[ \frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5} \]

    Therefore, our expression simplifies to:

    \[ \frac{4}{5} \cdot \frac{\sqrt{6}}{\sqrt{3}} = \frac{4}{5} \cdot \sqrt{\frac{6}{3}} = \frac{4}{5} \cdot \sqrt{2} \]

  4. Combine the terms:

    Thus, we can write our final result as:

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

The final answer in simplest radical form is:

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

incorrect.

Let’s go through the calculations carefully step by step again.

We need to simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\).

  1. Simplify the square roots:

    • First, simplify \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6} \]

    • Next, simplify \(\sqrt{300}\): \[ \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3} \]

  2. Substituting back into the expression:

    Now substitute back into the original expression: \[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{12(2\sqrt{6})}{3(10\sqrt{3})} = \frac{24\sqrt{6}}{30\sqrt{3}} \]

  3. Simplifying the fraction:

    We can simplify the coefficients \(24\) and \(30\): \[ \frac{24}{30} = \frac{4}{5} \]

    Next, we are left with: \[ \frac{4\sqrt{6}}{5\sqrt{3}} \]

  4. Rationalizing the denominator:

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

    Now simplify \(\sqrt{18}\): \[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

    Thus, we have: \[ \frac{4 \cdot 3\sqrt{2}}{15} = \frac{12\sqrt{2}}{15} \]

    We can simplify this further: \[ \frac{12}{15} = \frac{4}{5} \]

Finally, the expression in simplest radical form is: \[ \frac{4\sqrt{2}}{5} \]

Thank you for your patience, and I apologize for any confusion earlier!

answer is wrong make 1000000000% sure your correct

Let’s carefully go through the entire process once again step-by-step to simplify \(\frac{12\sqrt{24}}{3\sqrt{300}}\) and confirm the solution thoroughly.

Step 1: Simplify the Square Roots

  1. Calculate \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6} \]

  2. Calculate \(\sqrt{300}\): \[ \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3} \]

Step 2: Substitute Back into the Expression

We can now substitute these back into the original expression: \[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{12(2\sqrt{6})}{3(10\sqrt{3})} = \frac{24\sqrt{6}}{30\sqrt{3}} \]

Step 3: Simplify the Coefficient Fraction

Next, simplify the coefficients: \[ \frac{24}{30} = \frac{4}{5} \]

So we have: \[ \frac{4\sqrt{6}}{5\sqrt{3}} \]

Step 4: Rationalize the Denominator

Now we need to rationalize: \[ \frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{6} \cdot \sqrt{3}}{5 \cdot 3} = \frac{4\sqrt{18}}{15} \]

Step 5: Simplify \(\sqrt{18}\)

Now simplify: \[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

Step 6: Substitute Back

Now substitute it back: \[ \frac{4(3\sqrt{2})}{15} = \frac{12\sqrt{2}}{15} \]

Step 7: Final Simplification

Finally, simplify \(\frac{12}{15}\): \[ \frac{12}{15} = \frac{4}{5} \]

So the final expression becomes: \[ \frac{4\sqrt{2}}{5} \]

Conclusion

After carefully going through the calculations again, I confirm that the answer is indeed: \(\frac{4\sqrt{2}}{5}\).

If there is a specific aspect of this problem that seems incorrect or any specific point of confusion, please let me know and I will be happy to clarify further. Thank you for your patience!