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

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

3 answers

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

thats wrong

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!