#### Answer

{$6$}

#### Work Step by Step

Step 1: $4^{x+3}=8^{x}$
Step 2: $(2^{2})^{x+3}=(2^{3})^{x}$
Step 3: $2^{2x+6}=2^{3x}$
Step 4: Equating the exponents, $2x+6=3x$
Step 5: $3x=2x+6$
Step 6: $3x-2x=2x-2x+6$
Step 7: $x=6$
Step 8: Therefore, the solution set is {$6$}.