There is a mistake in your simplification
It is (x^2+4)^(1/2) (3x+5)^(1/3) { 4(x^2+4) + 3x(3x+5)} This will give you the text answer.
I am trying to figure out how to factor out any common factors out of this polynomial: 4(x^{2}+4)^{3/2}(3x+5)^{1/3} + (3x+5)^{4/3}(x^{2}+4)^{1/2}3x
What I thought to do was to look for each base raised to the smallest exponent present in each term. Looking at the polynomial it looks like (3x+5)^{1/3}, and (x^{2}+4)^{1/2} are the bases raised to the lowest exponent because 3/2 > 1/2, and 4/3>1/3 therefore I tried to factor these out. When I try to do this I end up with 4(x^{2}+4)^{2/2}+(3x+5)^{3/3}3x[(3x+5)(x^{2}+4)], and when I multiply the two binomials in the brackets I get 3x^{3}+5x^{2}+12x+20.
The answer in the text is (3x+5)^{1/3}(x^{2}+4)^{1/2}(13x^{2}+15x+16). I don't see how they arrived at this, and I also don't see how you could arrive at the original polynomial through multiplication of the terms in the answer. If someone could tell me what I am doing wrong, and explain how to arrive at the correct answer I would really appreciate it. Thanks a lot, I love this site.
Thanks for the reply, however I still don't understand how the bases raised to the 3/2, and 4/3 are less than the bases raised to the 1/3 and 1/2 (or then why if the bases (x2+4)^3/2, and (3x+5)^4/3 aren't less than the others why are they the terms that are factored out).