Issue with math in physics problem?

[fightboy]

Ok so for the problem I'm mainly having trouble with setting up one of the equations.
The solutions manual jumped from:
ω_iz([(2/5m_ER_E²)/(2/5m_ER_E²) + (2/3m_debrisR_E²)]-1)
to:
ω_iz([m_E/(m_E + (5/3m_debris))]-1)

I placed the brackets to clearly separate the fraction from the -1. Anyways I'm having trouble seeing how the math is done to get from the first equation to the second equation, how was it simplified? If anyone is confused on how i wrote the problem please just ask! Thanks!

 

[Mark44]

Ok so for the problem I'm mainly having trouble with setting up one of the equations.
The solutions manual jumped from:
ω_iz([(2/5m_ER_E²)/(2/5m_ER_E²) + (2/3m_debrisR_E²)]-1)
to:
ω_iz([m_E/(m_E + (5/3m_debris))]-1)

I placed the brackets to clearly separate the fraction from the -1. Anyways I'm having trouble seeing how the math is done to get from the first equation to the second equation

These aren't equations - they are expressions.

Inside the parentheses you have (2/5m_ER_E²)/(2/5m_ER_E²), which is just 1.

The fractions you wrote are ambiguous, which doesn't change what I wrote above.
Is the numerator $\frac{2}{5}m_ER_E^2$
or is it $\frac{2}{5m_ER_E^2}$?

, how was it simplified? If anyone is confused on how i wrote the problem please just ask! Thanks!

 

[fightboy]

These aren't equations - they are expressions.

Inside the parentheses you have (2/5m_ER_E²)/(2/5m_ER_E²), which is just 1.

The fractions you wrote are ambiguous, which doesn't change what I wrote above.
Is the numerator $\frac{2}{5}m_ER_E^2$
or is it $\frac{2}{5m_ER_E^2}$?

The numerator is $\frac{2}{5}m_ER_E^2$. I get that $\frac{2}{5}m_ER_E^2$/$\frac{2}{5}m_ER_E^2$ is equal to 1, I am mainly confused on how (2/3m_debrisR_E²) became (5/3m_debris) or if the manual made a mistake.

 

[gopher_p]

If you put an extra set of parentheses in the original expression, the $R_E^2$s cancel and you can multiply everything by $\frac{5}{2}$ to get $$\frac{\frac{2}{5}m_ER_E^2}{\frac{2}{5}m_ER_E^2+\frac{2}{3}m_{\text{debris}} R_E^2}=\frac{m_E}{m_E+\frac{5}{3}m_{\text{debris}}}$$