$$20736^{\frac{1}{8}}$$ Step 1: Find the prime factorization of 20736. $$20736 = 2^{10} \times 3^4$$ Step 2: Rewrite the expression using the prime factorization. $$(2^{10} \times 3^4)^{\frac{1}{8}}$$ Step 3: Apply the exponent rule $(a^m b^n)^p = a^{mp} b^{np}$. $$2^{10 \times \frac{1}{8}} \times 3^{4 \times \frac{1}{8}}$$ Step 4: Simplify the exponents. $$2^{\frac{10}{8}} \times 3^{\frac{4}{8}}$$ $$2^{\frac{5}{4}} \times 3^{\frac{1}{2}}$$ Step 5: Rewrite the expression using radicals. $$2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}} = 2 \times 2^{\frac{1}{4}} = 2\sqrt[4]{2}$$ $$3^{\frac{1}{2}} = \sqrt{3}$$ Step 6: Combine the terms. $$2\sqrt[4]{2} \times \sqrt{3}$$ Alternatively, we can look for a number that, when raised to the power of 8, equals 20736. Let $x = 20736^{\frac{1}{8}}$. Then $x^8 = 20736$. We can test small integer values. $1^8 = 1$ $2^8 = 256$ $3^8 = 6561$ $4^8 = 65536$ Since $3^8 < 20736 < 4^8$, the answer is not an integer. Let's re-examine the prime factorization: $20736 = 2^{10} \times 3^4$ $20736^{\frac{1}{8}} = (2^{10} \times 3^4)^{\frac{1}{8}} = 2^{\frac{10}{8}} \times 3^{\frac{4}{8}} = 2^{\frac{5}{4}} \times 3^{\frac{1}{2}}$ $2^{\frac{5}{4}} = 2^{1.25}$ $3^{\frac{1}{2}} = \sqrt{3} \approx 1.732$ $2^{1.25} = 2 \times 2^{0.25} = 2 \times \sqrt[4]{2}$ $\sqrt[4]{2} \approx 1.189$ $2 \times 1.189 \approx 2.378$ $2.378 \times 1.732 \approx 4.119$ Let's check if there was a simpler way to factor 20736. $20736 = 144^2 = (12^2)^2 = 12^4 = (2^2 \times 3)^4 = 2^8 \times 3^4$. Ah, there was a mistake in the initial prime factorization. Let's correct the prime factorization: $20736 = 144 \times 144$ $144 = 12 \times 12 = (2^2 \times 3) \times (2^2 \times 3) = 2^4 \times 3^2$ So, $20736 = (2^4 \times 3^2) \times (2^4 \times 3^2) = 2^8 \times 3^4$. Now, let's recalculate: $$20736^{\frac{1}{8}} = (2^8 \times 3^4)^{\frac{1}{8}}$$ Step 3 (corrected): Apply the exponent rule $(a^m b^n)^p = a^{mp} b^{np}$. $$2^{8 \times \frac{1}{8}} \times 3^{4 \times \frac{1}{8}}$$ Step 4 (corrected): Simplify the exponents. $$2^1 \times 3^{\frac{4}{8}}$$ $$2 \times 3^{\frac{1}{2}}$$ Step 5 (corrected): Rewrite the expression using radicals. $$2 \times \sqrt{3}$$ The final answer is $\boxed{2\sqrt{3}}$.