Question

$x-\frac1x=4$ হলে প্রমাণ কর যে, $x^4+\frac1{x^4}=322$

Answer 1

দেওয়া আছে,
$x-\frac1x=4$

বা, $\left(x-\frac{1}{x}\right)^2=\left(4\right)^2$

বা, $x^2-2\cdot x\cdot\frac1x+\frac{1}{x^2}=16$

বা, $x^2-2+\frac{1}{x^2}=16$

বা, $x^2+\frac{1}{x^2}=16+2$

বা, $x^2+\frac{1}{x^2}=18$

বা, $\left(x^2+\frac{1}{x^2}\right)^2=\left(18\right)^2$

বা, $(x^2)^2+2\cdot x^2\cdot\frac1{x^2}+\left(\frac{1}{x^2}\right)^2$$=324$

বা, $x^4+2+\frac{1}{x^4}=324$

বা, $x^4+\frac{1}{x^4}=324-2$

$\therefore x^4+\frac{1}{x^4}=322$ [Proved]

Comments

Rules Applied :

• $(a-b)^2=a^2-2ab+b^2$
• $(a+b)^2=a^2+2ab+b^2$