Question 730422
{{{ x^8 - 20x^4 + 64 = 0 }}}
Let {{{ z^2 = x^8 }}}
and {{{ z = x^4 }}}
{{{ z^2 - 20z + 64 = 0 }}}
{{{ ( z - 4 )*( z - 16 ) }}} ( just figured it out )
{{{ z = 4 }}}
{{{ z = 16 }}}
so
{{{ x^4 = 4 }}}
{{{ x^2 = 2 }}}
{{{ x^2 = -2 }}}
and
{{{ x^4 = 16 }}}
{{{ x^2 = 4 }}}
{{{ x^2 = -4 }}}
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Solving for {{{ x }}},
(1) {{{ x = sqrt(2) }}}
(2) {{{ x = -sqrt(2) }}}
(3) {{{ x = sqrt(2)*i }}}
(4) {{{ x = -sqrt(2)*i }}}
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(5) {{{ x = 2 }}}
(6) {{{ x= -2 }}}
(7) {{{ x = 2i }}}
(8) {{{ x = -2i }}}
The 8 roots are labeled (1) - (8)
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I'll check one of them
(4) {{{ x = -sqrt(-2) }}}
{{{ x^8 - 20x^4 + 64 = 0 }}}
{{{ ( -sqrt(-2)  )^8 - 20*( -sqrt(-2) )^4 + 64 = 0 }}}
{{{ 16*i^8 - 20*4*i^4 + 64 = 0 }}}
{{{ 16*(-1)^4 - 80*(-1)^2 + 64 = 0 }}}
{{{ 16 - 80 + 64 = 0 }}}
{{{ 0 = 0 }}}
OK
You can check the rest