How do you properly distribute powers in a fraction?

[ride5150]

im doing my traffic analysis homework and i don't understand how they went from "1" to "2" in the picture below. more specifically i don't know if I am taking the derivative wrong or if they are (those who wrote the solution). if you distribute the "k" that's at the beginning of the equation, you will get:

q = uf (k - [(k²))/k_j]^3.5)=0

then if i distribute the 3.5 power, i should get:

q = uf (k-[(k^5.5)/(kj^3.5)] = 0

then if i take dq/dk of this, shouldn't i get:

dq/dk= uf(1-[(5.5k^4.5)/(kj^3.5)]

?

the pic below is a screenshot of the solution, and theyre getting dq/dk to be:

dq/dk= uf(1-[(4.5k^3.5)/(kj^3.5)]

are they wrong?

 

[dynamicsolo]

Keep in mind that (a^b)^c = a^bc . So (k²)^3.5 is k⁷.

 

[ride5150]

Keep in mind that (a^b)^c = a^bc . So (k²)^3.5 is k⁷.

oh yeah that's right, how did they get 4.5k^3.5 then? wouldn't it be 7k⁶?

 

[dynamicsolo]

There seems to be some inconsistency between what you've typed and what you show in the attachment. Is the first equation you wrote supposed to have that "k²" in it?

Also, the first line in your attachment (what you call (1) ) is not an equation. Where does "q" come from? This isn't any way to get from (1) to (2) in what you displayed...

 

[ride5150]

There seems to be some inconsistency between what you've typed and what you show in the attachment. Is the first equation you wrote supposed to have that "k²" in it?

Also, the first line in your attachment (what you call (1) ) is not an equation. Where does "q" come from? This isn't any way to get from (1) to (2) in what you displayed...

sorry about the first post, that was completely wrong. i added the powers instead of multiplying them.

"equation" (1) is what "q" is equal to. there should be a "q=" in front of it, i accidentally cut it off while cropping the screenshot. the second equation is a derivative of the first (1) with respect to "k."

 

[dynamicsolo]

All right, I think this is sorted out now. There should not be a "k²" in that first equation and the asterisk in (1) of your attachment is a multiplication sign.

The first equation in your original post should read $q = k \cdot u_{f} [ 1 - (\frac{k}{k_{j}})^{3.5} ]$, so $$q = k \cdot u_{f} [ 1 - (\frac{k^{3.5}}{k_{j}^{3.5}}) ] = u_{f} [ k - (\frac{k^{4.5}}{k_{j}^{3.5}}) ].$$ I think you'll find at this point that the differential equation for dq/dk is correct. (I'm assuming k_j is a constant; it is not necessary for q to equal zero in the first line -- it could equal any real constant for the result in (2) to give dq/dk = 0 .)

 

[ride5150]

All right, I think this is sorted out now. There should not be a "k²" in that first equation and the asterisk in (1) of your attachment is a multiplication sign.

The first equation in your original post should read $q = k \cdot u_{f} [ 1 - (\frac{k}{k_{j}})^{3.5} ]$, so $$q = k \cdot u_{f} [ 1 - (\frac{k^{3.5}}{k_{j}^{3.5}}) ] = u_{f} [ k - (\frac{k^{<b>4.5</b>}}{k_{j}^{3.5}}) ]$$. I think you'll find at this point that the differential equation for dq/dk is correct. (I'm assuming k_j is a constant; it is not necessary for q to equal zero in the first line -- it could equal any real constant for the result in (2) to give dq/dk = 0 .)

ahh now i get it. you can't distribute the "k" before distributing the 3.5 power to the fraction (k/kj).

thank you for your help, i guess i need to refresh distributive laws:P