There's no one way to solve these sorts of problems, so I'm just going to guide you through my own problem solving process for this. The first thing I notice about this fraction
[tex] \frac{4^6\cdot9^5+6^9\cdot120}{8^4\cdot3^{12}-6^{11}} [/tex]
is that it's extremely messy. I want to clean it up as much as possible, which means I'd love to be able to factor something out from the top and the bottom of this fraction to make my life easier. Ideally something as big as possible, so I can do all of my simplifying in one nice step.
Now, what makes us able to factor numbers out? Take an expression like
[tex] 4\cdot3+4\cdot5 [/tex]
Why can we rewrite this as [tex] 4\cdot(3+5) [/tex] ? Well, think about what that first expression meant. We took three 4's and added five more 4's, so naturally, that should get us a total of 3 + 5 = 8 4's, which is exactly what that second expression captures.
What we want to find in this problem is some number that plays the same role 4 did in the above example. One way we can accomplish this is by breaking our numbers down completely, into their prime factorizations, and seeing where that leads.
The composite numbers we have in this problem are 4, 9, 6, 120, and 8. Taking the prime factorizations of each, we find that
4 = 2²
6= 2 · 3
8 = 2³
9 = 3²
120 = 2³ · 3 · 5
Subbing those back into our fraction, we get
[tex] \frac{4^6\cdot9^5+6^9\cdot120}{8^4\cdot3^{12}-6^{11}}=\frac{(2^2)^6\cdot(3^2)^5+(2\cdot3)^9\cdot(2^3\cdot3\cdot5)}{(2^3)^4\cdot3^{12}-(2\cdot3)^{11}}\\\\=\frac{2^{12}\cdot3^{10}+2^{9}\cdot3^9\cdot2^3\cdot3\cdot5}{2^{12}\cdot3^{12}-2^{11}\cdot3^{11}} \\\\=\frac{2^{12}\cdot3^{10}+2^{12}\cdot3^{10}\cdot5}{2^{12}\cdot3^{12}-2^{11}\cdot3^{11}} [/tex]
On the top, we see the number [tex] 2^{12}\cdot3^{10} [/tex] attached to both terms, so we can pull that out, and on the bottom, we have [tex] 2^{11}\cdot3^{11} [/tex]. Pulling these out, we get
[tex] \frac{2^{12}\cdot3^{10}+2^{12}\cdot3^{10}\cdot5}{2^{12}\cdot3^{12}-2^{11}\cdot3^{11}} = \frac{2^{12}\cdot3^{10}(1+5)}{2^{11}\cdot3^{11}(2\cdot3-1)}=\frac{2(6)}{3(5)}=\frac{2(2)}{5}=\frac{4}{5} [/tex]
So, 4/5 is our result.
