We need to simplify the expression $(\frac{216}{343})^{\frac{2}{3}}$.

1. Problem Description

We need to simplify the expression $(\frac{216}{343})^{\frac{2}{3}}$.

2. Solution Steps

First, we can rewrite the expression using the property $(a/b)^n = a^n / b^n$:

$(\frac{216}{343})^{\frac{2}{3}} = \frac{216^{\frac{2}{3}}}{343^{\frac{2}{3}}}$

Next, we can rewrite the fractional exponent as a radical:

$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

So, we have:

$\frac{216^{\frac{2}{3}}}{343^{\frac{2}{3}}} = \frac{(\sqrt[3]{216})^2}{(\sqrt[3]{343})^2}$

Now, we find the cube roots of 216 and 343:

$\sqrt[3]{216} = 6$ since $6*6*6 = 216$

$\sqrt[3]{343} = 7$ since $7*7*7 = 343$

Substitute these values back into the expression:

$\frac{(\sqrt[3]{216})^2}{(\sqrt[3]{343})^2} = \frac{6^2}{7^2}$

Calculate the squares:

$6^2 = 36$

$7^2 = 49$

So, the expression simplifies to:

$\frac{36}{49}$

3. Final Answer

$\frac{36}{49}$