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(12sqrt24)/(18sqrt32)=

$\frac{12 \sqrt{24}}{18 \sqrt{32}}$ =

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Simplify radical expressions involving fractions

Full solution

Q.

\frac{12 \sqrt{24}}{18 \sqrt{32}}

=

Apply Quotient Rule of Radicals: Apply the quotient rule of radicals to simplify the expression. $\newline$ $(12\sqrt{24})/(18\sqrt{32}) = (12/18) \times (\sqrt{24}/\sqrt{32})$
Simplify Fraction $12$ / $18$ : Simplify the fraction $\frac{12}{18}$ by dividing both numerator and denominator by their greatest common divisor, which is $6$ . $\left(\frac{12}{18}\right) = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
Find Prime Factorization: Simplify the radical expression $\sqrt{24}/\sqrt{32}$ by finding the prime factorization of the numbers under the radicals. $\newline$ $\sqrt{24} = \sqrt{(2 \times 2 \times 2 \times 3)} = \sqrt{2^3 \times 3}$ $\newline$ $\sqrt{32} = \sqrt{(2 \times 2 \times 2 \times 2 \times 2)} = \sqrt{2^5}$
Take Out Pairs of Prime Factors: Simplify the radicals by taking out pairs of prime factors. $\newline$ $\sqrt{24} = \sqrt{2^3 \times 3} = 2\sqrt{2 \times 3} = 2\sqrt{6}$ $\newline$ $\sqrt{32} = \sqrt{2^5} = 2^2\sqrt{2} = 4\sqrt{2}$
Substitute Simplified Radicals: Now, substitute the simplified radicals back into the expression. $(\frac{2}{3}) \times (\frac{2\sqrt{6}}{4\sqrt{2}})$
Divide Coefficients and Radicals: Simplify the expression by dividing the coefficients $\frac{2}{4}$ and the radicals $\frac{\sqrt{6}}{\sqrt{2}}$ . $\frac{2}{4} = \frac{1}{2}$ $\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3}$
Combine Coefficients and Radicals: Combine the simplified coefficients and radicals. $\newline$ $(\frac{1}{2}) \times (\frac{2}{3}) \times \sqrt{3} = (\frac{2}{6}) \times \sqrt{3} = (\frac{1}{3}) \times \sqrt{3}$
Final Simplified Form: The final simplified form of the expression is $(\frac{1}{3}) \times \sqrt{3}$ .

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