John claims that 3^2/3 * 3^3 to the 6th root = 3^2/3 * 3^3/6 = 3^6/18 = cubed root of 3
John's claim is incorrect.
To simplify the expression 3^2/3 * 3^3, we can use the property of exponentials that states when multiplying exponential expressions with the same base, we add the exponents.
So, 3^2/3 * 3^3 = 3^(2/3 + 3) = 3^(2/3 + 9/3) = 3^(11/3).
Now, let's simplify further. We can rewrite 3^(11/3) as (3^11)^(1/3), which means taking the 6th root.
Therefore, 3^2/3 * 3^3 = (3^11)^(1/3) = (3^11)^(1/6) = 3^(11/6).
In conclusion, the correct simplified expression of 3^2/3 * 3^3 to the 6th root is 3^(11/6), not the cube root of 3.