Let's start from the truth table for this exercise.
B1 B0 | A1 A0 | Y3 Y2 Y1 Y0 | B * A = Y
-------+-------+-------------+-----------
0 0 | 0 0 | 0 0 0 0 | 0 0 0
0 0 | 0 1 | 0 0 0 0 | 0 1 0
0 0 | 1 0 | 0 0 0 0 | 0 2 0
0 0 | 1 1 | 0 0 0 0 | 0 3 0
0 1 | 0 0 | 0 0 0 0 | 1 0 0
0 1 | 0 1 | 0 0 0 1 | 1 1 1
0 1 | 1 0 | 0 0 1 0 | 1 2 2
0 1 | 1 1 | 0 0 1 1 | 1 3 3
1 0 | 0 0 | 0 0 0 0 | 2 0 0
1 0 | 0 1 | 0 0 1 0 | 2 1 2
1 0 | 1 0 | 0 1 0 0 | 2 2 4
1 0 | 1 1 | 0 1 1 0 | 2 3 6
1 1 | 0 0 | 0 0 0 0 | 3 0 0
1 1 | 0 1 | 0 0 1 1 | 3 1 3
1 1 | 1 0 | 0 1 1 0 | 3 2 6
1 1 | 1 1 | 1 0 0 1 | 3 3 9
-------+-------+-------------+-----------
Resulting in the following 'raw' logical expressions: (Note: I'm using '*' to mean 'and', '+' to mean 'or', and single-quote to mean 'not'.)
Y3 = (B1 *B0 *A1 *A0 )
Y2 = (B1 *B0'*A1 *A0') + (B1 *B0'*A1 *A0 ) + (B1 *B0 *A1 *A0')
Y1 = (B1'*B0 *A1 *A0') + (B1'*B0 *A1 *A0 ) + (B1 *B0'*A1'*A0 ) + (B1 *B0'*A1 *A0 ) + (B1 *B0 *A1'*A0 ) + (B1 *B0 *A1 *A0')
Y0 = (B1'*B0 *A1'*A0 ) + (B1'+B0 *A1 *A0 ) + (B1 *B0 *A1'*A0 ) + (B1 *B0 *A1 *A0 )
The expressions above can be simplified (except for Y3), as the following K-maps show:
Y3 |A1'*A0'|A1'*A0 |A1 *A0 |A1 *A0'|
-------+-------+-------+-------+-------+
B1'*B0'| | | | |
-------+-------+-------+-------+-------+
B1'*B0 | | | | |
-------+-------+-------+-------+-------+
B1 *B0 | | | 1 | |
-------+-------+-------+-------+-------+
B1 *B0'| | | | |
-------+-------+-------+-------+-------+
Y3 = (B1 *B0 *A1 *A0 )
Y2 |A1'*A0'|A1'*A0 |A1 *A0 |A1 *A0'|
-------+-------+-------+-------+-------+
B1'*B0'| | | | |
-------+-------+-------+-------+-------+
B1'*B0 | | | | |
-------+-------+-------+-------+-------+
B1 *B0 | | | | 1 |
-------+-------+-------+-------+-------+
B1 *B0'| | | 1 | 1 |
-------+-------+-------+-------+-------+
Y2 = (B1 * A1*A0') + (B1 *B0'*A1 )
Y1 |A1'*A0'|A1'*A0 |A1 *A0 |A1 *A0'|
-------+-------+-------+-------+-------+
B1'*B0'| | | | |
-------+-------+-------+-------+-------+
B1'*B0 | | | 1 | 1 |
-------+-------+-------+-------+-------+
B1 *B0 | | | | 1 |
-------+-------+-------+-------+-------+
B1 *B0'| | 1 | 1 | |
-------+-------+-------+-------+-------+
Y1 = (B1'*B0 *A1 ) + ( B0 *A1 *A0') + (B1 *B0'*A1 )
Y0 |A1'*A0'|A1'*A0 |A1 *A0 |A1 *A0'|
-------+-------+-------+-------+-------+
B1'*B0'| | | | |
-------+-------+-------+-------+-------+
B1'*B0 | | 1 | 1 | |
-------+-------+-------+-------+-------+
B1 *B0 | | 1 | 1 | |
-------+-------+-------+-------+-------+
B1 *B0'| | | | |
-------+-------+-------+-------+-------+
Y0 = ( B0 * A0 )
So, the final, simplified expressions are:
Y3 = (B1 *B0 *A1 *A0 )
Y2 = (B1 *A1*A0') + (B1 *B0'*A1 )
Y1 = (B1'*B0 *A1 ) + (B0 *A1 *A0') + (B1 *B0'*A1 )
Y0 = (B0 *A0 )
Unfortunately, I don't have a convenient way of drawing the circuit(s) for this medium, but you'd need 1 4-input AND gate, 5 3-input AND gates, 1 2-input AND gate, 1 3-input OR gate, 1 2-input OR gate, and 3 NOT gates.
