Simplify the Boolean expression using K-Map: F(A,B,C,D)=∑m(1,3,4,5,9,11,14,15) + d∑(2,6,7,8).

Sure, let's simplify the given Boolean expression using Karnaugh Map (K-Map).

Step 1: Draw a 4-variable K-Map. The variables A, B, C, D are represented along rows and columns as follows:

CD\AB  00  01  11  10
00     m0  m1  m3  m2
01     m4  m5  m7  m6
11     m12 m13 m15 m14
10     m8  m9  m11 m10

Step 2: Mark the min terms (∑m) and don't care conditions (d∑) on the K-Map. The min terms are 1, 3, 4, 5, 9, 11, 14, 15 and the don't care conditions are 2, 6, 7, 8.

CD\AB  00  01  11  10
00     -   1   1   d
01     1   1   d   d
11     d   1   1   1
10     1   1   d   -

Step 3: Group the 1's and don't care conditions (if it helps to make a larger group). The groups should be as large as possible and can overlap. They should be rectangular and the number of cells in a group should be a power of 2.

Group 1: m1, m3, m5, m7 (vertical group in column 01 and 11)
Group 2: m4, m5, m12, m13 (horizontal group in row 01 and 11)
Group 3: m9, m11, m13, m15 (vertical group in column 01 and 11)
Group 4: m14, m15 (horizontal group in row 11)

Step 4: Write the simplified Boolean expression for each group.

Group 1: B'C
Group 2: AD
Group 3: BC
Group 4: CD

Step 5: Combine all the simplified expressions using OR operation.

So, the simplified Boolean expression is: F(A,B,C,D) = B'C + AD + BC + CD.