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Putting all with a base of 2,

2^10 * 2^40 * 2^120 = 2^n

Add up exponents, n = 170

$2^{10} \cdot 4^{20} \cdot 8^{40} = 2^n$

$2^{10} \cdot \left(2^2 \right)^{20} \cdot \left(2^3 \right)^{40} = 2^n$

$2^{10} \cdot 2^{40} \cdot 2^{120} = 2^n$

$2^{10 + 40 + 120} = 2^n$

$n = 10 + 40 + 120$

$n = \boxed{170}$