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As in the given question ${2^8} + {2^{10}} + {2^n}$ we have to find the value of n for which it is a perfect square.

For this we have to try to make it like ${a^2} + {b^2} + 2ab$

Hence, ${\left( {{2^4}} \right)^2} + {\left( {{2^5}} \right)^2} + {2^n}$

As for we have change it as ${2^n} = 2ab$

So that is equal to

${\left( {{2^4}} \right)^2} + {\left( {{2^5}} \right)^2} + {2.2^{n - 1}}$

So form this we know that as $a = {2^4},b = {2^5}$ ,

hence

${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$

${({2^4} + {2^5})^2}$ = \[{2^8} + {2^{10}} + {2.2^{4 + 5}}\]

Hence by comparing with real equation

\[{2^8} + {2^{10}} + {2.2^{4 + 5}}\] = ${\left( {{2^4}} \right)^2} + {\left( {{2^5}} \right)^2} + {2^n}$

From this \[{2^8},{2^{10}}\] is cancel out from both side the remaining term is

${2^{10}} = {2^n}$

By comparing the power we get $n = 10$

Hence $n = 10$ will be the answer for making this as perfect square

Whenever we have to give question like this just try to make it as ${\left( {a + b} \right)^2}$ as in the question that the power of $2$ is $8, 10$ it is multiply of $2$ or written as ${({2^4})^2}, {({2^5})^2}$ and n is unknown.

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