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Question Number 44444 by peter frank last updated on 29/Sep/18

simplify (√((4x^2 y)^(2/3) +(8x^2 y^2 )^4 ))

$${simplify}\:\:\:\:\sqrt{\left(\mathrm{4}{x}^{\mathrm{2}} {y}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\left(\mathrm{8}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Sep/18

(√((2^2 x^2 y)^(2/3) +(2^3 x^2 y^2 )^4 )) =(√(2^(4/3) x^(4/3) y^(2/3) +2^(12) x^8 y^8 )) =(√(2^(4/3) x^(4/3) y^(2/3) (1+2^((32)/3) x^((20)/3) y^((22)/3) ))) =2^(2/3) x^(2/3) y^(1/3) ×(√(1+2^((32)/3) x^((20)/3) y^((22)/3) ))

$$\sqrt{\left(\mathrm{2}^{\mathrm{2}} {x}^{\mathrm{2}} {y}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\left(\mathrm{2}^{\mathrm{3}} {x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$=\sqrt{\mathrm{2}^{\frac{\mathrm{4}}{\mathrm{3}}} {x}^{\frac{\mathrm{4}}{\mathrm{3}}} {y}^{\frac{\mathrm{2}}{\mathrm{3}}} +\mathrm{2}^{\mathrm{12}} {x}^{\mathrm{8}} {y}^{\mathrm{8}} } \\ $$$$=\sqrt{\mathrm{2}^{\frac{\mathrm{4}}{\mathrm{3}}} {x}^{\frac{\mathrm{4}}{\mathrm{3}}} {y}^{\frac{\mathrm{2}}{\mathrm{3}}} \left(\mathrm{1}+\mathrm{2}^{\frac{\mathrm{32}}{\mathrm{3}}} {x}^{\frac{\mathrm{20}}{\mathrm{3}}} {y}^{\frac{\mathrm{22}}{\mathrm{3}}} \right)} \\ $$$$=\mathrm{2}^{\frac{\mathrm{2}}{\mathrm{3}}} {x}^{\frac{\mathrm{2}}{\mathrm{3}}} {y}^{\frac{\mathrm{1}}{\mathrm{3}}} ×\sqrt{\mathrm{1}+\mathrm{2}^{\frac{\mathrm{32}}{\mathrm{3}}} {x}^{\frac{\mathrm{20}}{\mathrm{3}}} {y}^{\frac{\mathrm{22}}{\mathrm{3}}} } \\ $$

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