$\frac{(\sqrt{243})^{\sqrt{12}}}{3^{\frac{5}{2}} \cdot 2 \sqrt{3}}$

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\frac{(\sqrt{243})^{\sqrt{12}}}{3^{\frac{5}{2}} \cdot 2 \sqrt{3}}

Find Prime Factorization: Find the prime factorization of $243$ and simplify $\sqrt{243}$ . $243$ is $3^5$ , so $\sqrt{243}$ is $\sqrt{3^5}$ .
Simplify Square Root: Simplify $\sqrt{3^5}$ by taking out pairs of $3$ s. $\sqrt{3^5} = 3^2 \times \sqrt{3} = 9\sqrt{3}$ .
Simplify Another Square Root: Simplify $\sqrt{12}$ by finding its prime factorization. $\sqrt{12}$ is $\sqrt{4\times3}$ , which simplifies to $2\sqrt{3}$ .
Raise to Power: Raise $9\sqrt{3}$ to the power of $2\sqrt{3}$ . $(9\sqrt{3})^{2\sqrt{3}} = 9^{2\sqrt{3}} \times (\sqrt{3})^{2\sqrt{3}}$ .
Simplify Exponent: Simplify $3^{\frac{5}{2}}$ by finding its square root and then raising it to the power of $\frac{5}{2}$ . $3^{\frac{5}{2}} = (\sqrt{3})^5$ .
Combine Terms: Combine the denominator terms $3^{\frac{5}{2}}$ and $2\sqrt{3}$ . $3^{\frac{5}{2}} \times 2\sqrt{3} = (\sqrt{3})^5 \times 2\sqrt{3}.$
Simplify Denominator: Simplify the denominator $(\sqrt{3})^5 \times 2\sqrt{3}$ by combining like terms. $(\sqrt{3})^5 \times 2\sqrt{3} = 2 \times 3^{5/2} \times \sqrt{3}$ .
Divide Numerator: Divide the numerator by the denominator. $(9^{2\sqrt{3}} \cdot (\sqrt{3})^{2\sqrt{3}}) / (2 \cdot 3^{5/2} \cdot \sqrt{3})$ .
Cancel Common Terms: Simplify the expression by canceling out common terms. The $\sqrt{3}$ terms cancel out, and we're left with $\frac{9^{2\sqrt{3}}}{2 \cdot 3^{\frac{5}{2}}}$ .
Apply Power Rule: Realize that $9$ is $3^2$ and rewrite the expression. $(3^2)^{(2\sqrt{3})} / (2 \cdot 3^{\frac{5}{2}})$ .
Combine Powers: Apply the power of a power rule to the numerator. $(3^2)^{2\sqrt{3}} = 3^{4\sqrt{3}}$ .
Subtract Exponents: Combine the powers of $3$ in the numerator and denominator. $3^{4\sqrt{3}} / (2 \times 3^{5/2})$ .
Subtract Exponents: Combine the powers of $3$ in the numerator and denominator. $\frac{3^{4\sqrt{3}}}{2 \times 3^{\frac{5}{2}}}.$ Subtract the exponents of $3$ in the numerator and the denominator. $\frac{3^{4\sqrt{3} - \frac{5}{2}}}{2}.$