To simplify, notice that `x^2 +8xy + 16y^2` is a perfect square trinomial and can be factored as:``

`x^2 +8xy +16y^2 = (x+4y)^2`

So, you can rewrite the expression as:

`((x+4y)^2)^(1/3) * (x+4y)^(1/3)`

Then, you can simplify it using the property of exponent:

`(a^n)^m = a^(mn)`

`a =x+4y`

`n = 2`

`m =1/3`

So, `((x+4y)^2)^(1/3) = (x+4y)^(2/3)`

You now have: `(x+4y)^(2/3) * (x+4y)^(1/3)`

Notice that the two terms have the same base (x+4y). So you can use, `a^n * a^m = a^(m+n)`

`a=x+4y`

`n =2/3`

`m = 1/3`

So, `(x+4y)^(2/3+1/3) = (x+4y)^1`

*`2/3+1/3 =1`

Thus, the answer is `x+4y`

You can check your answer by assuming values for x and y. Then, plug-in in the original expression and the final answer. They should give the same value.

Say, x = 2 and y =3.

`(x^2 +8xy +16y^2)^(1/3) * (x+4y)^(1/3)`

`x+4y`

`2 + 4*3 = 14`

`(x^2+8xy+16y^2)^(1/3)(x+4y)^(1/3)=`

`=[(x+4y)^2)]^(1/3)(x+4y)^(1/3)=` `=[(x+4y)^2(x+4y)]^(1/3)=`

`=[(x+4y)^3]^(1/3)=` `x+4y`