This problem is a good review of laws of exponents.
First, we given two equations:
R = 3^81
R^R = 3^S
And it asks us to figure out an expression for S. Well, first of all, substitute the first equation into each "R" of the second equation
3^S = (3^81)^(3^81)
By the law of exponents that says (a^m)^n = a^(m*n), we can simplify this to:
3^S = (3^81)^(3^81) = 3^[81*(3^81)]
Both sides are now of the form 3 to the power of something, so we can equate the exponents.
S = 81*(3^81)
None of the answers are in that form. I notice, though, that 81 itself is a power of 3: 81 = 3^4. I'll substitute that into the first 81 of the expression for S
S = 81*(3^81) = (3^4)*(3^81)
Now, we can use another law of exponents that says (a^m)*(a^n) = a^(m+n)
S = 81*(3^81) = (3^4)*(3^81) = 3^(4+81) = 3^85
Answer = D, which coincides with the OA.
This was a really tricky question. Does all this make sense? Please let me know if there are any questions.
Mike
