Answer:

[tex]\boxed{\bf \: n = 6 \:} \\ [/tex]

Step-by-step explanation:

Given that,

[tex]\sf \: \dfrac{2 \times {2}^{2} \times {2}^{3} \times ... \times {2}^{n} }{ {2}^{n} } = 32768 \\ [/tex]

can be rewritten as

[tex]\sf \: \dfrac{{2}^{1 + 2 + 3 + ... + n} }{ {2}^{n} } = 32768 \\ [/tex]

can be further rewritten as

[tex]\sf \: \dfrac{{2}^{ \frac{n(n + 1)}{2} } }{ {2}^{n} } = 32768 \\ [/tex]

[tex]\sf \: {2}^{ \frac{n(n + 1)}{2} - n } = 32768 \\ [/tex]

[tex]\sf \: {2}^{ \frac{ {n}^{2} + n - 2n}{2}} = 32768 \\ [/tex]

[tex]\sf \: {2}^{ \frac{ {n}^{2} - n}{2}} = 32768 \\ [/tex]

[tex]\sf \: {2}^{ \frac{ {n}^{2} - n}{2}} = {2}^{15} \\ [/tex]

[tex]\sf \: \dfrac{ {n}^{2} - n}{2} = 15 \\ [/tex]

[tex]\sf \: {n}^{2} - n = 30 \\ [/tex]

[tex]\sf \: {n}^{2} - n - 30 = 0\\ [/tex]

[tex]\sf \: {n}^{2} - 6n + 5n - 30 = 0\\ [/tex]

[tex]\sf \: n(n - 6) + 5(n - 6) = 0\\ [/tex]

[tex]\sf \: (n - 6)(n + 5) = 0\\ [/tex]

[tex]\implies\sf \: n = 6 \: \: or \: \: n = - 5 \: \{rejected \} \\ [/tex]

[tex]\implies\boxed{\bf \: n = 6 \:} \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Formulae Used:

[tex]\sf \: {x}^{m} \times {x}^{n} = {x}^{m + n} \\ [/tex]

[tex]\sf \: {x}^{m} \div {x}^{n} = {x}^{m - n} \\ [/tex]

[tex]\sf \: 1 + 2 + 3 + ... + n = \dfrac{n(n + 1)}{2} \\ [/tex]