Question

simplifying complex fractions
simplify \\(\frac{\frac{\frac{16m^2}{m^2 + 5}}{4m}}{3m^2 + 15}\\).
options: 4m, 7m, 12m, 36m

Explanation:

Step1: Simplify the complex fraction

Recall that dividing by a fraction is multiplying by its reciprocal. So, \(\frac{\frac{\frac{16m^2}{m^2 + 5}}{4m}}{3m^2 + 15}=\frac{16m^2}{(m^2 + 5)\times4m}\div(3m^2 + 15)\)
First, simplify \(\frac{16m^2}{4m(m^2 + 5)}=\frac{4m}{m^2 + 5}\)
Then, factor \(3m^2 + 15 = 3(m^2 + 5)\)
Now, the expression becomes \(\frac{4m}{m^2 + 5}\div3(m^2 + 5)=\frac{4m}{m^2 + 5}\times\frac{1}{3(m^2 + 5)}\)? Wait, no, wait. Wait, the original complex fraction is \(\frac{\frac{\frac{16m^2}{m^2+5}}{4m}}{3m^2 + 15}\). Let's re - express the complex fraction step by step.
The numerator of the main fraction is \(\frac{\frac{16m^2}{m^2 + 5}}{4m}\), which is equal to \(\frac{16m^2}{(m^2 + 5)\times4m}=\frac{4m}{m^2 + 5}\) (dividing \(16m^2\) by \(4m\) gives \(4m\)).
The denominator of the main fraction is \(3m^2+15 = 3(m^2 + 5)\)
So the main fraction is \(\frac{\frac{4m}{m^2 + 5}}{3(m^2 + 5)}=\frac{4m}{m^2 + 5}\times\frac{1}{3(m^2 + 5)}\)? No, that's wrong. Wait, no, the complex fraction is \(\frac{\text{numerator}}{\text{denominator}}\) where numerator is \(\frac{\frac{16m^2}{m^2 + 5}}{4m}\) and denominator is \(3m^2 + 15\).
Wait, actually, \(\frac{\frac{A}{B}}{C}=\frac{A}{B\times C}\). So here, \(A=\frac{16m^2}{m^2 + 5}\), \(B = 4m\), \(C=3m^2 + 15\)
So \(\frac{\frac{\frac{16m^2}{m^2 + 5}}{4m}}{3m^2+15}=\frac{16m^2}{(m^2 + 5)\times4m\times(3m^2 + 15)}\)
Factor \(3m^2 + 15=3(m^2 + 5)\)
So substitute: \(\frac{16m^2}{4m(m^2 + 5)\times3(m^2 + 5)}=\frac{16m^2}{12m(m^2 + 5)^2}\)
Simplify \(16m^2\div12m=\frac{4m}{3}\)? Wait, this is not matching the options. Wait, maybe I misread the problem. Wait, the problem is \(\frac{\frac{\frac{16m^2}{m^2 + 5}}{4m}}{3m^2+15}\)? Or is it \(\frac{\frac{16m^2}{m^2 + 5}}{\frac{4m}{3m^2 + 15}}\)? Oh! Maybe the original problem is a complex fraction where the numerator is \(\frac{16m^2}{m^2 + 5}\) and the denominator is \(\frac{4m}{3m^2 + 15}\). That would make more sense. Let's assume that.
So if the complex fraction is \(\frac{\frac{16m^2}{m^2 + 5}}{\frac{4m}{3m^2 + 15}}\), then we use the rule \(\frac{a/b}{c/d}=\frac{a\times d}{b\times c}\)
So \(a = 16m^2\), \(b=m^2 + 5\), \(c = 4m\), \(d=3m^2 + 15\)
Then \(\frac{16m^2\times(3m^2 + 15)}{(m^2 + 5)\times4m}\)
Factor \(3m^2 + 15=3(m^2 + 5)\)
Substitute: \(\frac{16m^2\times3(m^2 + 5)}{(m^2 + 5)\times4m}\)
Cancel out \((m^2 + 5)\) from numerator and denominator.
Then we have \(\frac{48m^2}{4m}\) (since \(16\times3 = 48\))
Simplify \(\frac{48m^2}{4m}=12m\) (dividing \(48m^2\) by \(4m\): \(48\div4 = 12\) and \(m^2\div m=m\))

Step2: Match with the options

We got the result as \(12m\), which is one of the options.

Answer:

12m (corresponding to the option "12m")