And you i just need someone to correct my work from my answer to the answer my teacher wrote i'm confused and need to find where i messed up

See below.

Step-by-step explanation:

I'll work it out myself,  then we should be able to see the mistake:-

6 ∛ (64x^5y^9)

= 6 * 4 * x^(5/3) * y^3

= 24 x^(3/3) * x^(2/3) y^3

= 24 x y^3 ∛(x^2)

You've got confused withe the cube root  (∛) on the first line. You've interpreted it   as ' 3  times the square root' so you were on the wrong road from the beginning.

Note that y^9 has cube root y^(9/ 3) = y^3  and x^5 has cube root of x^(5/3).   The square root of x^9 is x^(9/2).

answer:

your answer is: c. 196 7/8 in^3.