jmc

The numerator is equal to $x^{1+2+3+\cdots +15}$. The exponent is the sum of the first 15 consecutive positive integers, so its sum is $\frac{15\cdot 16}{2}=120$. So the numerator is $x^{120}$.

The denominator is equal to $x^{2+4+6+\cdots +20}=x^{2(1+2+3+\cdots +10)}$. The exponent is twice the sum of the first 10 consecutive positive integers, so its sum is $2\cdot \frac{10\cdot 11}{2}=110$. So the denominator is $x^{110}$.

The entire fraction becomes $\frac{x^{120}}{x^{110}}=x^{120-110}=x^{10}$. Plugging in $x=2$ yields $2^{10}=\boxed{1024}$.