To solve the cubic equation \( 27x^3 - 54x^2 + 36x - 8 = 0 \), we can first look for rational roots using the Rational Root Theorem, which suggests testing the factors of the constant term (-8) over the factors of the leading coefficient (27).

The potential rational roots could be \( \pm 1, \pm 2, \pm 4, \pm 8 \) for the numerator and \( \pm 1, \pm 3, \pm 9, \pm 27 \) for the denominator. Thus, we can test the following possible rational roots: \( \pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm \frac{2}{3}, \pm \frac{2}{9}, \pm \frac{4}{3}, \pm \frac{8}{27} \).

Let's start testing these values:

1. Testing \( x = 1 \):
\[
27(1)^3 - 54(1)^2 + 36(1) - 8 = 27 - 54 + 36 - 8 = 1 \quad (\text{not a root})
\]

2. Testing \( x = \frac{2}{3} \):
\[
27\left(\frac{2}{3}\right)^3 - 54\left(\frac{2}{3}\right)^2 + 36\left(\frac{2}{3}\right) - 8
\]
\[
= 27\left(\frac{8}{27}\right) - 54\left(\frac{4}{9}\right) + 36\left(\frac{2}{3}\right) - 8
\]
\[
= 8 - 24 + 24 - 8 = 0 \quad (\text{this is a root})
\]

Since \( x = \frac{2}{3} \) is a root, we can use synthetic division or polynomial long division to factor the cubic polynomial:

Dividing \( 27x^3 - 54x^2 + 36x - 8 \) by \( 3x - 2 \):

### Synthetic Division by \( 3x - 2 \)

Coefficients: \( 27, -54, 36, -8 \)

We are dividing by \( \frac{2}{3} \):

1. Bring down the \( 27 \).
2. Multiply \( 27 \) by \( \frac{2}{3} \) to get \( 18 \) and add to \( -54 \): \( -54 + 18 = -36 \).
3. Multiply \( -36 \) by \( \frac{2}{3} \) to get \( -24 \) and add to \( 36 \): \( 36 - 24 = 12 \).
4. Multiply \( 12 \) by \( \frac{2}{3} \) to get \( 8 \) and add to \( -8 \): \( -8 + 8 = 0 \).

The result of the synthetic division is:
\[
27x^2 - 36x + 12.
\]

Now we factor \( 27x^2 - 36x + 12 \):

Factor out 3:
\[
3(9x^2 - 12x + 4).
\]

Now we can find the roots using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
a = 9, b = -12, c = 4.
\]
\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(9)(4)}}{2(9)} = \frac{12 \pm \sqrt{144 - 144}}{18} = \frac{12}{18} = \frac{2}{3}.
\]

Since the discriminant is zero, we have a double root. The roots of the cubic equation \( 27x^3 - 54x^2 + 36x - 8 \) are:
\[
x = \frac{2}{3} \quad \text{(triple root)}.
\]

Thus, the solutions to the equation \( 27x^3 - 54x^2 + 36x - 8 = 0 \) are:
\[
x = \frac{2}{3}.
\]