If , , are cube roots of -8 , then what is ^2 p^2+ ^2 q^2+ ^2 r^2 ^2 p^2+ ^2 q^2+ ^2 r^2 equal to ?

Step-by-step solution

Since , , are the cube roots of -8 , they are the roots of the equation z^3 + 8 = 0 . Thus, we have ^3 = ^3 = ^3 = -8 . The product of the roots of the cubic equation z^3 + 8 = 0 is given by = -8 . Using these properties, we can find relations between the roots: ^3 = ^2 = ^3 = ^2 = ^3 = ^2 = Let the given expression be E = ^2 p^2+ ^2 q^2+ ^2 r^2 ^2 p^2+ ^2 q^2+ ^2 r^2 . Consider multiplying the denominator by the fraction : ( ^2 p^2 + ^2 q^2 + ^2 r^2) = p^2 + ^3 q^2 + ^2 r^2 Now, substitute the derived relations in