The total amount of the invoice is $564.00, which includes the tax. So 116% is equivalent to $564.00.
Therefore 100%=100×564/116=100×141/29=$486.21 to the nearest cent. This is the amount before tax is added. So the tax must be 564-486.21=$77.79.
This is the same as 564-100×564/116=564(1-100/116)=564×16/116=564×4/29=2256/29=$77.79, which confirms the answer you were given.