To simplify this expression, we can first combine the terms with the same base raised to different exponents.

We have:
[tex]0.6x^{\frac{1}{7}} \cdot 27^{\frac{2}{7}} \cdot 25^{\frac{4}{7}}[/tex]

Now, we know that:
[tex]27 = 3^3[/tex]
[tex]25 = 5^2[/tex]

Substitute these values into our expression:
[tex]0.6x^{\frac{1}{7}} \cdot (3^3)^{\frac{2}{7}} \cdot (5^2)^{\frac{4}{7}}[/tex]

Simplify the exponents:
[tex]0.6x^{\frac{1}{7}} \cdot 3^{\frac{6}{7}} \cdot 5^{\frac{8}{7}}[/tex]

Now, multiply the coefficients and combine the terms with the same base:
[tex]0.6 \cdot 3^{\frac{6}{7}} \cdot 5^{\frac{8}{7}} \cdot x^{\frac{1}{7}}[/tex]

Therefore, the simplified expression is:
[tex]0.6 \cdot 3^{\frac{6}{7}} \cdot 5^{\frac{8}{7}} \cdot x^{\frac{1}{7}}[/tex]