<s>[INST] Easy exponents question I have the GRE Friday... I got hung up on this easy exponents problem (I think it was these exponents, don't recall exactly)
$$\frac{6^{14}}{2^7 \times 3^5} = ? $$
The answer is $2^73^9$, but could anyone double check for me?
 [/INST] Essentially yes. You need to use the rules $(xy)^a = x^a y^a$ and also $x^ax^b = x^{a+b}$ (the latter of which sometimes looks like $\frac{x^a}{x^b} = x^a x^{-b} = x^{a-b}$ for division). Here you first factor $6$ and then simplify. Explicitly:
$$
\frac{6^{14}}{2^7\cdot 3^5} = \frac{(2\cdot3)^{14}}{2^7\cdot 3^5} = \frac{2^{14}\cdot3^{14}}{2^7\cdot 3^5} = 2^{14-7}\cdot3^{14-5} = 2^7\cdot 3^{9}.
$$
Good luck on the GRE!
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