Alan

"Hey, how much is 8 to the power of 0? And why?"

Perhaps this will help:

To put a little flesh on Guest#1'sanswer, and assuming the question is 

"When b^2 is written in base c, the result is 121_c. When c^2 is written in base b, the result is 71_b."

we have:

As follows

In the first part of the question the filled in box for BC is incorrect!

B = [-4, 2

        0, 5]

C = [0, 4, -1

        1, 0, 4]

B*C = [-4*0+2*1, -4*4+2*0,  -4*(-1)+2*4

           0*0+5*1,  0*4+5*0,    0*(-1)+5*4]

B*C = [2, -16, 12

           5,  0,   20]

For the second part, A - B means subtract corresponding elements in A and B.

I hope the following helps:

To be clear, when I wrote "There are an infinite number of possible values..."  I meant "There are an infinite number of possible values for r ..."

Just one careless mistake Melody, it should be \(-\frac{325}{e}+120\)

1/9 = 3^-2 

27 = 3³ 

So we have   3^-2 = (3³)^(2x+1)     or  3^-2 = 3^3*(2x+1)

We can equate powers:   -2 = 3*(2x + 1)

Can you take it from here?

"Can an infinite set be placed into a one-to-one correspondence?"

Yes.  For example:

1, 2, 3, 4, 5, .....

|   |   |   |   |   .....

2, 4, 6, 8, 10, ....

The natural numbers are in one-to-one correspondence with the positive even integers.

A^-1v  rotates v clockwise by 36degrees to a.

B^-1v  reflects to d.

C^-1v  goes to b, which seems to be identical to v.

D^-1v  shrinks v to c.

Substitute n = 1 into each of a,b,c and d.  It is clear that a, b and c then resut in a number which is not -39.  On the other hand, d does result in -39, so d is the right answer.