Help intuiting an algebraic manipulation:
(6 + x)^n/6^n = (1 + x/6)^n

Question

Help intuiting an algebraic manipulation:
(6 + x)^n/6^n = (1 + x/6)^n

barrycarter · Answer

Well, (6+x)/6 = 1 + x/6

datanewb · Answer

I have verified that the following equations are equal:

$$\frac{(6 + x)^n}{6^n} = (1 + \frac{x}{6})^n $$
 for n = 0,1,2.  And I assume that is equal for all values of n.  In the solutions to some homework, a similar fraction was treated this way, and it then made the fraction easier to work on.  My question is, how does one see that fractions of this sort can be modified from the representation on the left to the representation on the right?  And, vice-versa?

anonymous · Answer

$$\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$$

anonymous · Answer

$$(1+\frac{x}{6})^n=(\frac{1}{1}+\frac{x}{6})^n=(\frac{6}{6}+\frac{x}{6})^n=(\frac{6+x}{6})^n$$

datanewb · Answer

Building off of the answer given by @barrycarter and thinking about it a little further, I think I've figured out how to "intuit" that such a modification can work.

$$ \frac{x^n}{y^n} $$

lol, yes, @satellite73 , that is what I was just about to type, thank you!

datanewb · Answer

I get it now, combing the answers of @henpen and @satellite73.  Thank you all for answering!  The step that my mind was missing was the step satellite wrote.