Question

ঘন-এর সাহায্যে সরল কর: $(m+n)^6$$-(m-n)^6$$-12mn(m^2-n^2)^2$

Answer 1

$(m+n)^6$$-(m-n)^6$$-12mn(m^2-n^2)^2$

$=(m+n)^6-(m-n)^6$$-12mn\left\lbrace\left(m+n\right)(m-n)\right\rbrace^2$

$=\left\lbrace(m+n)^2\right\rbrace^3$$-\left\lbrace(m-n)^2\right\rbrace^3$$-3\times\left(m+n\right)^2(m-n)^2\times4mn$

ধরি,
$(m+n)^2=p$ এবং $(m-n)^2$

এখন,
$p-q$

$=(m+n)^2-(m-n)^2$

$=(m^2+2mn+n^2)-(m^2-2mn+n^2)$

$=m^2+2mn+n^2-m^2+2mn-n^2$

$=4mn$

তাহলে প্রদত্ত রাশি দাঁড়ায়,
$p^3-q^3-3pq(p-q)$

$=(p-q)^3$

$=(4mn)^3$

$=64m^3n^3$ [Answer]