1xx2x4x2x6543x6x54x32x567x6543x25x7x6543x5 = P; P tetrated to P = W; w(P) = answer

Steps to Solve

1. Identify and Multiply the Numbers in the Expression

First, let's clarify the expression given. It looks like we need to multiply several numbers: $$ P = 1 \times 2 \times 4 \times 2 \times 6543 \times 6 \times 54 \times 32 \times 567 \times 6543 \times 25 \times 7 \times 6543 \times 5 $$

2. Calculate P

Next, multiply these numbers together:

• Start with multiplying the smaller numbers first for simplicity.
• Calculate ( 1 \times 2 = 2 )
• Then, ( 2 \times 4 = 8 )
• Now multiply ( 8 \times 2 = 16 )
• Continue with ( 16 \times 6543 = 104688 )
• Now ( 104688 \times 6 = 628128 )
• Multiply ( 628128 \times 54 = 3397872 )
• Next, ( 3397872 \times 32 = 108355584 )
• Continue with ( 108355584 \times 567 = 61418682368 )
• Multiply ( 61418682368 \times 6543 = 402909096557824 )
• Then ( 402909096557824 \times 25 = 10072727413945600 )
• Now ( 10072727413945600 \times 7 = 70409091897519200 )
• Finally, ( 70409091897519200 \times 6543 \times 5 ) gives us the final product for P.

3. Calculate the Final Product P

Continue multiplying:

• First multiply ( 70409091897519200 \times 6543 = 460904912295920896000 )
• Then multiply by 5: $$ P = 460904912295920896000 \times 5 = 2304524561479604480000 $$

4. Tetrate P to get W

Now we need to tetrate ( P ) to itself, ( W = P \uparrow P ). Tetration involves repeated exponentiation: $$ W = P^{P} $$

However, this number is so large it's generally left in terms of its expression rather than calculated.

5. Find w(P)

For ( w(P) ), we can represent it as: $$ w(P) = W $$