Solve the Nested Division Expression: 38:(6:(2:6))

First, let's write the multiplication exercise in the inner parentheses as a fraction:

$38:(6:\frac{2}{6})=$

Now let's flip the fraction to create a multiplication exercise:

$38:(6\times\frac{6}{2})=$

Let's add the 6 to the fraction's numerator to create a multiplication exercise:

$38:\frac{6\times6}{2}=$

Let's flip the fraction to create a multiplication exercise:

$38\times\frac{2}{6\times6}=$

Let's add the 38 to the fraction's numerator to create a multiplication exercise:

$\frac{38\times2}{6\times6}=$

Let's break down the 38 and the two 6's in the denominator into smaller multiplication exercises:

$\frac{19\times2\times2}{2\times3\times2\times3}=$

Let's reduce the 2 in both numerator and denominator:

$\frac{19}{3\times3}=\frac{19}{9}$

Let's separate the fraction into an addition exercise between fractions:

$\frac{18+1}{9}=\frac{18}{9}+\frac{1}{9}=2+\frac{1}{9}=2\frac{1}{9}$