20736 power 1/8

$$2^{\frac{10}{8}} \times 3^{\frac{4}{8}}$$ $$2^{\frac{5}{4}} \times 3^{\frac{1}{2}}$$

$$2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}} = 2 \times 2^{\frac{1}{4}} = 2\sqrt[4]{2}$$ $$3^{\frac{1}{2}} = \sqrt{3}$$

$$2\sqrt[4]{2} \times \sqrt{3}$$

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 $2\sqrt{3}$.