Writing without negative exponents

[stellardrone]

Hello! This is my first post, I hope this is in the right sub-section/topic. Currently working through an Algebra book (Introduction):
I have difficulties with a double fraction.

[MATH] \frac{2x^{-3}}{x^{5}} [/MATH] should be written without negative exponent.
I am aware that [MATH] \frac{x^{-3}}{x^{5}} [/MATH] can be written as [MATH] x^{-3-5} [/MATH] and this yields [MATH] \frac{1}{x^{8}} [/MATH].

Another way is to write it like this:

[MATH]\frac{2*\frac{1}{x^3}}{x^{5}} [/MATH]
According to the solution this results in [MATH]\frac{2}{\dfrac{x^{3}}{x^{5}}} = \frac{2}{x^{3} * x^{5}} = \frac{2}{x^{8}} [/MATH]




These last two steps confuse me. Why can you write this in such a way under each other?

I always multiplied double-fractions by the inverse; in my understanding this would be 2/x^3 * x^5 /1 = 2x^5 / x^3 , but this seems to be wrong.

I do not know, how to get the same result (2/x^8) using the long way.
Where is my reasoning error?

 

[skeeter]

[MATH]\dfrac{\left(\frac{2}{x^3}\right)}{x^5} = \dfrac{2}{x^3} \div x^5 = \dfrac{2}{x^3} \div \dfrac{x^5}{1} = \dfrac{2}{x^3} \cdot \dfrac{1}{x^5} = \dfrac{2}{x^8}[/MATH]

 

[Dr.Peterson]

According to the solution this results in [MATH]\frac{2}{\dfrac{x^{3}}{x^{5}}} = \frac{2}{x^{3} * x^{5}} = \frac{2}{x^{8}} [/MATH]

You wrote the compound fraction incorrectly; it should have been this, where the lower fraction bar is larger:
[MATH]\dfrac{\frac{2}{x^{3}}}{x^{5}} = \frac{2}{x^{3} x^{5}} = \frac{2}{x^{8}}[/MATH]

 

[stellardrone]

[MATH]\dfrac{\left(\frac{2}{x^3}\right)}{x^5} = \dfrac{2}{x^3} \div x^5 = \dfrac{2}{x^3} \div \dfrac{x^5}{1} = \dfrac{2}{x^3} \cdot \dfrac{1}{x^5} = \dfrac{2}{x^8}[/MATH]

Thank you @skeeter, that makes perfect sense. Writing it that way makes it easy to understand.

 

[stellardrone]

T

You wrote the compound fraction incorrectly; it should have been this, where the lower fraction bar is larger:
[MATH]\dfrac{\frac{2}{x^{3}}}{x^{5}} = \frac{2}{x^{3} x^{5}} = \frac{2}{x^{8}}[/MATH]

Thank you @Dr.Peterson for bringing this to my attention.
What's the exact difference? Would my incorrectly written fraction mean 2/1 : x^3/x^5?

 

[HallsofIvy]

\(\displaystyle \frac{a}{\frac{b}{c}}\) means "a divided by the fraction b/c". To divide by a fraction you "invert and multiply" so that is \(\displaystyle a\left(\frac{c}{b}\right)= \frac{ac}{b}\).

\(\displaystyle \frac{\frac{a}{b}}{c}\) means "the fraction a/b divided by c". That is \(\displaystyle \frac{a}{bc}\).

 

[stellardrone]

Thank you @HallsofIvy. Got it now!

 

[Steven G]

What's the exact difference? Would my incorrectly written fraction mean 2/1 : x^3/x^5?

Consider these two fraction. The main division line (the longer one) makes a big difference!

[MATH]\dfrac{\dfrac{100}{10}}{2} = \dfrac{10}{2} = 5[/MATH]
While [MATH]\dfrac{100}{\dfrac{10}{2}}= \dfrac{100}{5}= 20[/MATH]

 

[stellardrone]

Thank you @Jomo. I can see the difference.
(On a sidenote: Do all of you use the LaTeX Math Mode in your posts? Or is there a better way?)