જો ${7 \over {{2^{1/2}} + {2^{1/4}} + 1}}$$ = A + B{.2^{1/4}} + C{.2^{1/2}} + D{.2^{3/4}}$, તો $A+B+C+D= . . .$
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d
(d) \({7 \over {{2^{1/2}} + {2^{1/4}} + 1}} = {{7\,.\,({2^{1/4}} - 1)} \over {({2^{1/4}} - 1)\,[{{({2^{1/4}})}^2} + {2^{1/4}}.1 + {1^2}]}}\)
(d) \({7 \over {{2^{1/2}} + {2^{1/4}} + 1}} = {{7\,.\,({2^{1/4}} - 1)} \over {({2^{1/4}} - 1)\,[{{({2^{1/4}})}^2} + {2^{1/4}}.1 + {1^2}]}}\)
\(= {{7\,.\,({2^{1/4}} - 1)} \over {{2^{3/4}} - 1}} = A + B\,.\,{2^{1/4}} + C.\,{2^{1/2}} + D{.2^{3/4}}\)
==> \(7\,.\,{2^{1/4}} - 7 = (A - D)\,{2^{3/4}} + (2B - A) + (2C - B){.2^{1/4}}\)\( + (2D - C){2^{1/2}}\)
==> \((2B - A + 7) + (A - D){2^{3/4}} + (2C - B - 7){2^{1/4}}\) \(+ (2D - C){2^{1/2}} = 0\)
==> \(2B - A + 7 = A - D = 2C - B - 7 = 2D - C = 0\)
==> \(A = D = 1,\,B = - 3,\,C = 2\).
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