I'll just ask a portion of the question:

How does the fractional exponent in (3/7)9^(7/3) simplify to 384/7 and how does (3/4)9^1/3 simplify to [I don't know the answer to this... it's either 12, 3/7, or 3/4]?

I'd understand it if the exponent was a square root, but with a cube root, I'm lost.

( \frac 3 7 ) 9^{7/3} does not equal \frac {384}{7}.

And:


( \frac 3 4) 9^ {1/3}


does not equal any of the possible answers you gave. Note that the cube root of 9 is an irrational number, so the answer can't be a simple fraction.

I'm confused now... my teacher and my textbook claim it's possible O_O

http://college.cengage.com/mathematics/blackboard/content/larson/calc8e/04/04e/se04e01087.html

I think I figured it out. You mis-wrote the equations - instead of 9 being taken to the power of 7/3 it should be 8 to the 7/3.

8^(7/3) = 8^((1/3) ^7)

8^(1/3) =2, and 2^7 = 128. Do you see why 8^(1/3) is 2?

3/7 x 128 = 384/7.

As for the second problem, you have (3/4)8^(1/3). From what I showed above can you complete this?

Oh, that was stupid. Thank you!