Question 119631
{{{x^6-64=0}}} Start with the given equation



{{{(x^2-4)(x^4+4x^2+16)=0}}} Factor the left side by using the difference of cubes formula



{{{(x+2)(x-2)(x^4+4x^2+16)=0}}} Factor {{{x^2-4}}} to get {{{(x+2)(x-2)}}}



Now set each factor equal to zero:


{{{x+2=0}}}, {{{x-2=0}}} or {{{x^4+4x^2+16=0}}}


Now solve for x for the first two factors:


{{{x=-2}}}, {{{x=2}}} or {{{x^4+4x^2+16=0}}}



So the first two roots are {{{x=-2}}} or {{{x=2}}}


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Now let's solve {{{x^4+4x^2+16=0}}}



Let {{{w=x^2}}}


{{{w^2+4w+16=0}}}



Let's use the quadratic formula to solve for w:



Starting with the general quadratic


{{{aw^2+bw+c=0}}}


the general solution using the quadratic equation is:


{{{w = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{w^2+4*w+16=0}}} ( notice {{{a=1}}}, {{{b=4}}}, and {{{c=16}}})





{{{w = (-4 +- sqrt( (4)^2-4*1*16 ))/(2*1)}}} Plug in a=1, b=4, and c=16




{{{w = (-4 +- sqrt( 16-4*1*16 ))/(2*1)}}} Square 4 to get 16  




{{{w = (-4 +- sqrt( 16+-64 ))/(2*1)}}} Multiply {{{-4*16*1}}} to get {{{-64}}}




{{{w = (-4 +- sqrt( -48 ))/(2*1)}}} Combine like terms in the radicand (everything under the square root)




{{{w = (-4 +- 4*i*sqrt(3))/(2*1)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{w = (-4 +- 4*i*sqrt(3))/(2)}}} Multiply 2 and 1 to get 2




After simplifying, the quadratic has roots of


{{{w=-2 + 2*sqrt(3)i}}} or {{{w=-2 - 2*sqrt(3)i}}}


Remember, {{{w=x^2}}}. So {{{x^2=-2 + 2*sqrt(3)i}}} or {{{x^2=-2 - 2*sqrt(3)i}}}



Now take the square root of both sides for each case


{{{x=0+-sqrt(-2 + 2*sqrt(3)i)}}} or {{{x=0+-sqrt(-2 - 2*sqrt(3)i)}}}


Now break up the plus/minus


{{{x=sqrt(-2 + 2*sqrt(3)i)}}}, {{{x=-sqrt(-2 + 2*sqrt(3)i)}}}, {{{x=sqrt(-2 - 2*sqrt(3)i)}}}, or {{{x=-sqrt(-2 - 2*sqrt(3)i)}}}


So the roots of {{{x^6-64=0}}} are 



{{{x=-2}}}, {{{x=2}}}, {{{x=sqrt(-2 + 2*sqrt(3)i)}}}, {{{x=-sqrt(-2 + 2*sqrt(3)i)}}} or {{{x=sqrt(-2 - 2*sqrt(3)i)}}}, or {{{x=-sqrt(-2 - 2*sqrt(3)i)}}}