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Question Number 96329 by  M±th+et+s last updated on 31/May/20

$$x\lfloor x\lfloor x\lfloor x\rfloor \rfloor \rfloor =88$$ $$x>0$$

Commented byprakash jain last updated on 31/May/20

$$88=2^3\times 11$$ $$x=n+f\Rightarrow n^4⩽88$$ $$3^4=81$$ $$\Rightarrow 3⩽n<4$$ $$x=3+f(f>0)$$ $$(3+f)\lfloor (3+f)\lfloor 9+3f\rfloor \rfloor =88$$ $$(3+f)\lfloor (3+f)(9+\lfloor 3f\rfloor )\rfloor =88$$ $$(3+f)\lfloor 27+9f+3\lfloor 3f\rfloor +f\lfloor 3f\rfloor \rfloor =88$$ $$81+27f+...=88$$ $$f<\frac{1}{3}\because 27\times \frac{1}{3}=9>7\Rightarrow \lfloor 3f\rfloor =0$$ $$(3+f)\lfloor 27+9f\rfloor =88$$ $$81+27f+f\lfloor 9f\rfloor +3\lfloor 9f\rfloor =88$$ $$27f+f\lfloor 9f\rfloor +3\lfloor 9f\rfloor =7$$ $$x<\frac{1}{3}\Rightarrow \lfloor 9f\rfloor =2,1,0$$ $$\lfloor 9f\rfloor =0\Rightarrow f=\frac{7}{27}\Rightarrow \lfloor 9f\rfloor >0\left(invalid\right)$$ $$\lfloor 9f\rfloor =1\Rightarrow f=\frac{1}{7}\Rightarrow \lfloor 9f\rfloor =1\left(valid\right)$$ $$\lfloor 9f\rfloor =2\Rightarrow f=\frac{1}{29}\Rightarrow \lfloor 9f\rfloor =0\left(invalid\right)$$ $$x=3+f=\frac{22}{7}$$

Commented by M±th+et+s last updated on 31/May/20

$$correctsolutionthankyou$$

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