Final answer:
You multiply the exponents when a power is raised to another power. The original expression (0.2) (x^{2/3})^{4/5} simplifies to x^{8/15} after applying this rule.
Step-by-step explanation:
To simplify the expression (0.2) (x^{2/3})^{4/5}, start by addressing the exponent raised to another exponent. According to the rules of exponents, when you raise a power to another power, you multiply the exponents. Here, we have x^{2/3} raised to the 4/5 power. The rule is similar to the simplification of (5^3)^4 = 5^{3\cdot4} = 5^{12} as explained in the references provided. Hence for our expression, we multiply 2/3 by 4/5 which gives us 2/3 \cdot 4/5 = 8/15. So, our expression simplifies to x^{8/15}.
The coefficient 0.2 is not attached to a variable and can be considered as part of the simplification process, but it does not affect the exponentiation of x. Therefore, the simplified form of the given expression is x^{8/15}, which matches answer choice (b).