Question

Factor completly x^8- 1/81. please help:)

Answer 1

Answer:

`{x}^(8) - (1)/(81) = ( {x}^(4) + (1)/(9) )( {x}^(2) + (1)/(3) )(x + (1)/( √(3) ) )(x - (1)/( √(3) ) )`

Explanation:

see the picture attached for further explanation.

Answer 2

`\large\boxed{\left(x^4+(1)/(9)\right)\left(x^2+(1)/(3)\right)\left(x+(\sqrt3)/(3)\right)\left(x-(\sqrt3)/(3)\right)}`

Explanation:

`Use\\\\a^2-b^2=(a-b)(a+b)\\\\(a^n)^m=a^(nm)\\\\x^8-(1)/(81)=x^(4\cdot2)-(1)/(9^2)=(x^4)^2-\left((1)/(9)\right)^2=\left(x^4+(1)/(9)\right)\underbrace{\left(x^4-(1)/(9)\right)}_((*))\\\\(*)=x^(2\cdot2)-(1)/(3^2)=(x^2)^2-\left((1)/(3)\right)^2=\left(x^2+(1)/(3)\right)\underbrace{\left(x^2-(1)/(3)\right)}_((**))\\\\(**)=x^2-(3)/(9)=x^2-((\sqrt3)^2)/(3^2)=x^2-\left((\sqrt3)/(3)\right)^2=\left(x+(\sqrt3)/(3)\right)\left(x-(\sqrt3)/(3)\right)`