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Question Number 191486 by MATHEMATICSAM last updated on 24/Apr/23

$$a\sqrt{a}+b\sqrt{b}=183\mathrm{and}b\sqrt{a}+a\sqrt{b}=182$$$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{value}\mathrm{of}\frac{9}{5}(a+b)?$$

Answered by mr W last updated on 24/Apr/23

$$letA=\sqrt{a}$$$$B=\sqrt{b}$$$${AB}^2+A^{2}B=182$$$$\Rightarrow AB(A+B)=182$$$$A^3+B^3=183$$$${(A+B)}^3-3AB(A+B)=183$$$${(A+B)}^3-3\times 182=183$$$$\Rightarrow A+B=\sqrt[3]{183+3\times 182}=9$$$$a+b=A^2+B^2={(A+B)}^2-2AB$$$$=9^2-2\times \frac{182}{9}=\frac{365}{9}$$$$\frac{9}{5}(a+b)=\frac{9}{5}\times \frac{365}{9}=73$$

Commented by mehdee42 last updated on 24/Apr/23

$$thatwasperfect$$

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