Question

Factor completely x^8-1/81

Answer 1

Remember the difference of 2 perfect squares
(a²-b²)=(a-b)(a+b)
1/81=(1/9)²
so
x⁸=(x⁴)²

x⁸-1/81=(x⁴)²-(1/9)²=(x⁴-1/9)(x⁴+1/9)
oh look, 1/9=(1/3)²
x⁴=(x²)²
we can factor that first one again
x⁴-1/9=(x²)²-(1/3)²=(x²-1/3)(x²+1/3)

so factored completely is (x²-1/3)(x²+1/3)(x⁴+1/9)
you could go further and force taht first term to be another difference by doing 1/3=(√(1/3))², but I wouldn't


(x²-1/3)(x²+1/3)(x⁴+1/9) is factored