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Question Number 145344 by mathdanisur last updated on 04/Jul/21

z^4 = 49 - 20(√6) ⇒ z=?

$$\boldsymbol{{z}}^{\mathrm{4}} \:=\:\mathrm{49}\:-\:\mathrm{20}\sqrt{\mathrm{6}}\:\:\Rightarrow\:\boldsymbol{{z}}=? \\ $$

Answered by liberty last updated on 04/Jul/21

z^2 = (√(49−2(√(600)))) =(√((25+24)−2(√(25×24)))) z^2 = (√(25))−(√(24)) = 5−2(√6) z =(√(5−2(√6))) =(√((3+2)−2(√(3×2)))) z = (√3)−(√2)

$$\:{z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{49}−\mathrm{2}\sqrt{\mathrm{600}}}\:=\sqrt{\left(\mathrm{25}+\mathrm{24}\right)−\mathrm{2}\sqrt{\mathrm{25}×\mathrm{24}}} \\ $$$${z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{25}}−\sqrt{\mathrm{24}}\:=\:\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}\: \\ $$$${z}\:=\sqrt{\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}}\:=\sqrt{\left(\mathrm{3}+\mathrm{2}\right)−\mathrm{2}\sqrt{\mathrm{3}×\mathrm{2}}} \\ $$$$\:{z}\:\:=\:\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\: \\ $$

Commented by mathdanisur last updated on 04/Jul/21

thank you Ser

$${thank}\:{you}\:{Ser} \\ $$

Answered by Olaf_Thorendsen last updated on 04/Jul/21

z^4 = 49−20(√6) z = ±((49−20(√6)))^(1/4) or ±((49−20(√6)))^(1/4) .i ∣z∣ = ((49−20(√6)))^(1/4) ∣z∣ = (((5−2(√6))^2 ))^(1/4) ∣z∣ = (√(5−2(√6))) ∣z∣ = (√(((√3)−(√2))^2 )) ∣z∣ = (√3)−(√2) Finally, z = ±((√3)−(√2)), ±((√3)−(√2))i

$${z}^{\mathrm{4}} \:=\:\mathrm{49}−\mathrm{20}\sqrt{\mathrm{6}} \\ $$$${z}\:=\:\pm\sqrt[{\mathrm{4}}]{\mathrm{49}−\mathrm{20}\sqrt{\mathrm{6}}}\:\mathrm{or}\:\pm\sqrt[{\mathrm{4}}]{\mathrm{49}−\mathrm{20}\sqrt{\mathrm{6}}}.{i} \\ $$$$ \\ $$$$\mid{z}\mid\:=\:\sqrt[{\mathrm{4}}]{\mathrm{49}−\mathrm{20}\sqrt{\mathrm{6}}} \\ $$$$\mid{z}\mid\:=\:\sqrt[{\mathrm{4}}]{\left(\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{2}} } \\ $$$$\mid{z}\mid\:=\:\sqrt{\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}} \\ $$$$\mid{z}\mid\:=\:\sqrt{\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$\mid{z}\mid\:=\:\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Finally},\:{z}\:=\:\pm\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right),\:\pm\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right){i} \\ $$

Commented by mathdanisur last updated on 04/Jul/21

Thank you Ser, answer: (√3)−(√2)

$${Thank}\:{you}\:{Ser},\:{answer}:\:\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}} \\ $$

Commented by mr W last updated on 04/Jul/21

if (√3)−(√2) is a root, then −((√3)−(√2)) is also a root, and ((√3)−(√2))i is also a root, and −((√3)−(√2))i is also a root.

$${if}\:\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\:{is}\:{a}\:{root},\:{then} \\ $$$$−\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right)\:{is}\:{also}\:{a}\:{root},\:{and} \\ $$$$\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right){i}\:{is}\:{also}\:{a}\:{root},\:{and} \\ $$$$−\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right){i}\:{is}\:{also}\:{a}\:{root}. \\ $$

Commented by mathdanisur last updated on 04/Jul/21

Thanks Sir

$${Thanks}\:{Sir} \\ $$

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