Understanding the evaluation of a fraction

Question

Sorry if this is a little simple to ask here; but I'm at a loss as to where else to look for help with this one.

I've been attempting to understand the evaluation of a fraction. The question I'm trying to answer begins with:

$$\frac{8}{27}^\frac{3}{2}$$

I believe that this is equivalent to:

$$(\frac{8}{27}^\frac{1}{2})^3$$

or

$$(\frac{8}{27}^3)^\frac{1}{2}$$

Proceeding from there, I managed to get to:

$$(\frac{512}{19683})^\frac{1}{2}$$

and

$$(\frac{\sqrt{8}}{3\sqrt{3}})^3$$

I wasn't able to see any sensible way of proceeding with either. After a lot of failed attempts, I checked the answer, which is given as:

$$\frac{16}{243}\sqrt{6}$$

Having looked at this, I considered that:

$(\frac{\sqrt{8}}{3\sqrt{3}})^3$ could be written as $(\frac{2\sqrt{2}}{3\sqrt{3}})^3$ which I believe is equivalent to $(\frac{2\sqrt{6}}{9})^3$

But here I hit a wall again - perhaps I'm going in the complete wrong direction with this, but I am wondering if anyone can't help me to understand how best to reach the correct solution.

Answer 1

We have $$\left(\frac{512}{19683}\right)^{\frac12}=\frac {\sqrt{512}}{\sqrt{19683}}=\frac{16\sqrt2}{81\sqrt3}=\frac {16\sqrt6}{243}$$

We could also have

$$\left(\frac{\sqrt8}{3\sqrt3}\right)^3=\left(\frac{2\sqrt2}{3\sqrt3}\right)^3=\frac {8\sqrt8}{27\sqrt{27}}=\frac{16\sqrt2}{81\sqrt3}=\frac {16\sqrt6}{243}$$

Answer 2

Since $8/27 = 2^3/3^3 = (2/3)^3$ we can see that $$((2/3)^3)^{3/2}=(2/3)^{9/2}=(2/3)^{8/2}(2/3)^{1/2} = (2/3)^4(\sqrt2/\sqrt{3})=2^4/3^4 (\sqrt6/3)=(2^4/3^5)\sqrt{6}=\frac{16\sqrt{6}}{243}$$