An algebra problem by Sakanksha Deo

First, we should know the identity.

$\large x^7 - y^7 = ( x - y ) ( x^6 + x^5 y + x^4 y^2 + x^3 y^3 + x^2 y^4 + x y^5 + y^6 )$

Now,

$\large \frac{1}{ \sqrt[7]{531441} + \sqrt[7]{472392} + \sqrt[7]{419904} + \sqrt[7]{373248} + \sqrt[7]{331776} + \sqrt[7]{294912} + \sqrt[7]{262144} }$

$= \large \frac{ 1 }{ \sqrt[7]{9^6} + \sqrt[7]{ 9^5 \times 8 } + \sqrt[7]{9^4 \times 8^2 } + \sqrt[7]{ 9^3 \times 8^3 } + \sqrt[7]{ 9^2 \times 8^4 } + \sqrt[7]{ 9 \times 8^5 } + \sqrt[7]{ 9^6 } }$

$= \large \frac{ 1 }{ \sqrt[7]{9^6} + \sqrt[7]{ 9^5 \times 8 } + \sqrt[7]{9^4 \times 8^2 } + \sqrt[7]{ 9^3 \times 8^3 } + \sqrt[7]{ 9^2 \times 8^4 } + \sqrt[7]{ 9 \times 8^5 } + \sqrt[7]{ 9^6 } } \times \frac{ ( \sqrt[7]{9} - \sqrt[7]{8} ) }{ ( \sqrt[7]{9} - \sqrt[7]{8} ) }$

$= \frac{ \sqrt[7]{9} - \sqrt[7]{8} }{ ( \sqrt[7]{9} )^7 - ( \sqrt[7]{8} )^7 }$

$= \frac{ \sqrt[7]{9} - \sqrt[7]{8} }{1} = \frac{ \sqrt[7]{a} - \sqrt[7]{b} + \sqrt[7]{c} }{d}$

Therefore,

$\large a - b + c + d = 9 - 8 + 0 + 1 = \boxed{2}$

It is a nice question. Please write 373248 in place of 373243 and also insert a '+' sign in the expression. It is simple question knowing factors of (a^7 - b^7).