* Simplify.

a to the power of 4/3 times c ^1/6 all over 9b^3..... everything is in the parenthesis and is raised to the power of -2....times the whole thing to,

a to the power of 3 times b to the power of two all over 3c.... everything is in the parenthesis and is raised to -1/4

******Is the answer (3^17/4)(b^11/2) all over (a^41/12)(c^1/12) ?

[ a^(4/3) c^(1/6) (1/9)b^-3 ]^-2

= [ a^(-8/3) c^(-1/3) (81) b^6 ]

I can not follow you after that, sorry

To simplify the given expression, let's break it down step by step:

1. Start by simplifying the expressions inside the parentheses which are raised to the power of -2:
a^(4/3) * c^(1/6) / 9b^3

2. Now, let's simplify the entire expression as a whole by raising it to the power of -2:
(a^(4/3) * c^(1/6) / 9b^3)^(-2)

3. To raise the expression to a negative power, we can invert the entire expression and change the sign of the exponent:
1 / (a^(4/3) * c^(1/6) / 9b^3)^(2)

4. Simplify the above expression by squaring each term inside the parentheses:
1 / ((a^(4/3))^2 * (c^(1/6))^2 / (9b^3)^2)

5. Simplify the exponents inside the parentheses by multiplying them:
1 / (a^(8/3) * c^(1/3) / (81b^6))

6. Next, simplify the expression inside the parentheses by dividing the exponents:
1 / (a^(8/3 - 1) * c^(1/3) / (81b^6))

7. Simplifying further:
1 / (a^(5/3) * c^(1/3) / (81b^6))

8. Now, simplify the second expression inside the parentheses which is raised to the power of -1/4:
(a^3 * b^2 / 3c)^(-1/4)

9. Similar to the previous steps, we can invert the expression and change the sign of the exponent:
1 / (a^3 * b^2 / 3c)^(1/4)

10. Simplify the expression by taking the fourth root of each term:
1 / (a^(3/4) * b^(1/2) / (3c)^(1/4))

11. Simplify the denominator:
1 / (a^(3/4) * b^(1/2) / (3^(1/4)c^(1/4)))

12. Lastly, multiply the simplified expressions inside the parentheses:
1 / (a^(3/4) * b^(1/2) / ((3c)^(1/4) * 3^(1/4)))

After following these steps, the answer will be in the form:
1 / (a^(3/4) * b^(1/2) / ((3c)^(1/4) * 3^(1/4)))

Please, double-check your calculations to ensure accuracy.