jmc

The largest perfect square less than 399 is $19^2=361$, and the largest perfect cube less than 399 is $7^3=343$. Any perfect fourth power is already a square, so we may skip to the largest fifth power less than $399$, which is $3^5=243$, Again, a sixth power is a square (and a cube), so we look to the largest seventh power less than $399$, which is $2^7=128.$ Eighth, ninth and tenth powers may be skipped again because they would already have been included as perfect squares or cubes, and there is no eleventh power less than $399$ other than $1$. Thus the largest perfect power less than 399 is $19^2=361$, and $a+b=19+2=\boxed{21}$.