The break down of a negative binomial equation

[mcanski]

Firstly, I want to note I'm a post college student who is attempting to teach himself calculus. I'm reading Calculus Made Easy by Silvanus P. Thompson and Martin Gardner, St. Martin's Press, 1998 ed.

My question comes from page 56 Case of a Negative Exponent

y + dy= (x + dx)^-2

= x^-2(1 + dx/x)^-2

I don't understand how the author got from (x + dx)^-2 to the answer x^-2(1 + dx/x)^-2

If someone could either breakdown the process, show me where to go and see examples of how this process is done, or point me in the direction to what emphasis of math I should read to better learn the process I will be grateful. Any help will be appreciated.

 

[Cyosis]

Try to take it from here:

$$(x+dx)^{-2}=\left(\frac{x}{x}(x+dx)\right)^{-2}$$

Realize that x/x is just 1 for x!=0 so you can put it in there without changing the expression.

 

[brennon80]

Cyosis, thanks for the next step. I however, am still quite frustrated with this. IMHO, leaving out obscure simplification steps for the reader to deduce on their own begins to miss the mark of making anything "simple". That said, I'm just not seeing the next step here. Is the x/x then getting multiplied into the binomial? I've tried that, but haven't made any headway. Would you help me to carry on from here?

Many thanks!

 

[Cyosis]

All that is being done is dividing the argument by x. This however changes the expression so you have to multiply by x so that the equality holds.

$$(x+dx)^{-2}=\left(\frac{x}{x}(x+dx)\right)^{-2}=\left(x \frac{x+dx}{x}\right)^{-2}$$

You should be able to carry out the division yourself.

 

[brennon80]

That helped tremendously--thank you!