How can I design a 4-to-16 decoder using two 3-to-8 decoders and 16 two-input AND gates?
you have to design a 4x16 decoder using two 3x8 decoders. here is the schematic that may help you.







simulate this circuit &ndash; Schematic created using CircuitLab

the two squares are two 3x8 decoders with enable lines. the three selection lines of each decoders are connected together as common line(X,Y,Z) , the enable lines are ACTIVE LOW, they are also connected together with a common line W , but the second one having a NOT gate connected within. So, there are now 4 selection inputs i.e W,X,Y,Z. For the values 0000 to 0111 ,the first decoder will turn on giving the decoded outputs 0 to 7 , and for 1000 to 1111 , the second decoder will turn on , giving decoded output 8 to 15. How? Because for the first 8 combinations, the W bit is 0 , so it is a 1 for the first decoder, and enable line is on(ACTIVE LOW) , but it goes through a NOT GATE and then to the ACTIVE LOW enable port of the second decoder, so it remains 0 , so the second decoder doesn't activate. then for the next 8 combinations, the W line is 1 , which becomes 0 inside the 1st decoder(ACTIVE LOW) and the 1st decoder turns off, but it goes through a NOT gate and then ACTIVE LOW enable port of 2nd decoder, becomes a 1 and activates the next outputs 8 to 15. so you get a total 16 outputs (0-15) 

If you use two 3x8 decoders you need not use 16 AND gates, and If you want to use AND gates only, then think about the internal circuitry of a decoder, think of this diagram , and you can easily construct using AND gates.