Question

find a natural no n, such that 2^8+2^10+2^n is a perfect square no

Answer 1

Answer:

Required natural number is 10.

Step-by-step explanation:

= > 2^8 + 2^10 + 2^n

= > ( 2^4 )^2 + ( 2^5 )^2 + 2{ 2^( n - 1 ) }

On observing the produced situation we can say that ( 2^4 )^2 + ( 2^5 )^2 + 2{ 2^( n - 1 ) } is given in the formula of a^2 + b^2 + 2ab that's a perfect square of ( a + b ).

Here, 2^4 is written in place of a and 2^5 is written in place of b. And the remaining terms are 2ab{ where a is 2^4 and b is 2^5 } and 2{ 2^( n - 1 ) }.

Both should be equal, since they are satisfying the situation.

Thus,

= > 2( 2^4 x 2^5 ) = 2{ 2^( n - 1 ) }

= > 2^4 x 2^5 = 2^( n - 1 )

= > 2^{ 4 + 5 } = 2^( n - 1 )

= > 2^9 = 2^( n - 1 )

Since bases are same and equal, powers must be equal.

= > 9 = n - 1

= > 9 + 1 = n

= > 10 = n

Hence the required natural number is 10.