Explanation of a simple exponent rule in a derivative problem needed

[AwesomeSN]

Explanation of a "simple" exponent rule in a derivative problem needed

After differentiation, I get this : 9(t-2/2t+1)⁸ * 5/(2t+1)²

Now this simplifies into 45(t-2)/(2t+1)¹⁰

Now, I am wondering what the property is that combines the exponent from the top with the bottom to get an power of 10 in the simplified form.

*Note. the t-2/2t+1 is t-2 over 2t+1, the 5/(2t+1)² is 5 over (2t+1)², and the 45(t-2)/(2t+1)¹⁰ is 45(t-2) over (2t+1)¹⁰.

 

[Pi-Bond]

From the rules of exponents:

$a^n a^m = a^{n+m}$

 

[ArcanaNoir]

if you start with $$9 (\frac{t-2}{2t+1})^8 * \frac{5}{(2t+1)^2}$$
then you can write $$9 \frac{(t-2)^8}{(2t+1)^8}* \frac{5}{(2t+1)^2}$$
and then you add the exponents when multiplying like bases, thus getting $$45 \frac{(t-2)^8}{(2t+1)^{10}}$$

Note that $$(2t+1)^8(2t+1)^2=(2t+1)^{10}$$

edit:
boo, pibond beat me cause I'm slow at latex! :P

 

[AwesomeSN]

Thank you. I was aware of the rule for exponents, but applying the power of 8 to both the numerator and denominator escaped me, now it all makes sense!