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Question Number 185422 by Ml last updated on 21/Jan/23

Answered by hmr last updated on 21/Jan/23

2(√5) = (√(20)) = (20)^(1/2) →a= ((2(√5)))^(1/5) = (((20)^(1/2) ))^(1/5) = (20)^(1/(10)) (a^(16) )^(1/3) = a^((16)/3) = ((20)^(1/(10)) )^((16)/3) = (20)^(8/(15)) = ((20^8 ))^(1/(15))

$$\mathrm{2}\sqrt{\mathrm{5}}\:=\:\sqrt{\mathrm{20}}\:=\:\left(\mathrm{20}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\rightarrow{a}=\:\sqrt[{\mathrm{5}}]{\mathrm{2}\sqrt{\mathrm{5}}}\:=\:\sqrt[{\mathrm{5}}]{\left(\mathrm{20}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} }\:=\:\left(\mathrm{20}\right)^{\frac{\mathrm{1}}{\mathrm{10}}} \\ $$$$ \\ $$$$\sqrt[{\mathrm{3}}]{{a}^{\mathrm{16}} }\:=\:{a}^{\frac{\mathrm{16}}{\mathrm{3}}} \:=\:\left(\left(\mathrm{20}\right)^{\frac{\mathrm{1}}{\mathrm{10}}} \right)^{\frac{\mathrm{16}}{\mathrm{3}}} =\:\left(\mathrm{20}\right)^{\frac{\mathrm{8}}{\mathrm{15}}} \\ $$$$=\:\sqrt[{\mathrm{15}}]{\mathrm{20}^{\mathrm{8}} } \\ $$

Commented by Frix last updated on 21/Jan/23

Typo in line 3

$$\mathrm{Typo}\:\mathrm{in}\:\mathrm{line}\:\mathrm{3} \\ $$

Commented by hmr last updated on 21/Jan/23

yes sir, thank you. I will edit it.

$${yes}\:{sir},\:{thank}\:{you}. \\ $$$${I}\:{will}\:{edit}\:{it}. \\ $$

Answered by Frix last updated on 21/Jan/23

a=2^(1/5) 5^(1/(10)) a^((16)/3) =2^((16)/(15)) 5^(8/(15)) =20^(8/(15))

$${a}=\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{5}}} \mathrm{5}^{\frac{\mathrm{1}}{\mathrm{10}}} \\ $$$${a}^{\frac{\mathrm{16}}{\mathrm{3}}} =\mathrm{2}^{\frac{\mathrm{16}}{\mathrm{15}}} \mathrm{5}^{\frac{\mathrm{8}}{\mathrm{15}}} =\mathrm{20}^{\frac{\mathrm{8}}{\mathrm{15}}} \\ $$

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