Question Number 166241 by MathsFan last updated on 15/Feb/22
$$\:\boldsymbol{{x}}^{\mathrm{5}} −\mathrm{1}=\mathrm{0} \\ $$$$\:\boldsymbol{{please}}\:\boldsymbol{{how}}\:\boldsymbol{{do}}\:\boldsymbol{{i}}\:\boldsymbol{{find}}\:\boldsymbol{{for}}\:\boldsymbol{{all}}\:\boldsymbol{{the}} \\ $$$$\:\boldsymbol{{values}}\:\boldsymbol{{of}}\:\boldsymbol{{x}}? \\ $$
Answered by batamatikayoga last updated on 16/Feb/22
$${x}^{\mathrm{5}} =\mathrm{1}={e}^{\mathrm{2}\pi{ik}} ,{k}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\} \\ $$$${x}={e}^{\frac{\mathrm{2}}{\mathrm{5}}\pi{ik}} \\ $$
Commented by MathsFan last updated on 16/Feb/22
$$\mathrm{1}+{e}^{\pi{i}} =\mathrm{0} \\ $$$$\mathrm{1}=−{e}^{\pi{i}} ???? \\ $$
Commented by alephzero last updated on 16/Feb/22
$${yes} \\ $$$$\mathrm{1}\:=\:−{e}^{{i}\pi} \\ $$$$\Rightarrow\:−\mathrm{1}\:=\:−\left(−{e}^{{i}\pi} \right)\:=\:{e}^{{i}\pi} \\ $$
Answered by JDamian last updated on 16/Feb/22
$${x}={e}^{\frac{\mathrm{2}\pi{i}}{\mathrm{5}}{k}} \:\:\:\forall{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\right\} \\ $$
Commented by MathsFan last updated on 16/Feb/22
$${thank}\:{you}\:{sir} \\ $$