Question

Find product using suitable identity:

$(x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})(x^4+\frac{1}{x^4})$

Answer 1

As we know that

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

$(x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})(x^4+\frac{1}{x^4})=(x^2-\frac{1}{x^2})(x^2+\frac{1}{x^2})(x^4+\frac{1}{x^4})$

$=((x^2{)}^2-(\frac{1}{x^2}{)}^2)(x^4+\frac{1}{x^4})=(x^4-\frac{1}{x^4})(x^4+\frac{1}{x^4})=(x^4{)}^2-(\frac{1}{x^4}{)}^2=x^8-\frac{1}{x^8}$