Why does ((4^x In 4)x^4 - 4^x 4x^3) / x^8 = (x^3 4^x (x In 4- 4)) / x^8)?

Question

Why does ((4^x In 4)x^4 - 4^x 4x^3) / x^8 = (x^3 4^x (x In 4- 4)) / x^8)?

 

It says that in my book and i don't understand how to get to that. Thank you very much guys

Answer 1

That is what you get when you factor the numerator of the expression on the left.  Note that the terms in the numerator on the left have a common factor of 4^x x³.  Hence, the numerator may be expressed as

 

4^x x³ (x ln4 - 4).

 

Now divide this by the denominator, which is x⁸, and you get 

 

4^x x³ (x ln4 - 4) / x⁸

 

which is the expression on the right.  This can be simplified further to 

 

4^x (x ln4 - 4) / x⁵.