Question Number 188482 by mathlove last updated on 02/Mar/23
$$\mathrm{512}{x}^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\mathrm{1} \\ $$$${find}\:\:{volue}\:\:{of}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left({x}^{\mathrm{2}} \right)^{{n}} =? \\ $$
Answered by mr W last updated on 02/Mar/23
$$\mathrm{512}{x}^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\mathrm{1} \\ $$$$\left(\mathrm{512}\right)^{−\mathrm{3}} \left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =\left(−\mathrm{1}\right)^{−\mathrm{3}} =−\mathrm{1} \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =−\mathrm{512}^{−\mathrm{3}} =\left(−\mathrm{8}\right)^{−\mathrm{9}} \\ $$$$\Rightarrow{x}^{−\mathrm{3}} =−\mathrm{8}=\left(−\mathrm{2}\right)^{\mathrm{3}} =\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{−\mathrm{3}} \\ $$$$\Rightarrow{x}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow{x}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{x}^{\mathrm{2}{n}} =\frac{\frac{\mathrm{1}}{\mathrm{4}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}}=\frac{\mathrm{1}}{\mathrm{3}}\:\checkmark \\ $$
Commented by mathlove last updated on 02/Mar/23
$${i}\:{do}'{nt}\:{understand}\:{for}\:{this} \\ $$$$\left(\mathrm{512}\right)^{−\mathrm{3}} \left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =−\left(\mathrm{512}\right)^{−\mathrm{3}} \\ $$$${pleas}\:{discrib}\:{this} \\ $$
Commented by mr W last updated on 02/Mar/23
$$\left(\mathrm{512}\right)^{−\mathrm{3}} \left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =\left(−\mathrm{1}\right)^{−\mathrm{3}} =−\mathrm{1}\:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\left(\mathrm{512}\right)^{\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{−\left(\mathrm{1}−{x}^{−\mathrm{3}} \right)} =\left(−\mathrm{1}\right)^{−\mathrm{3}} \left(\mathrm{512}\right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =−\left(\mathrm{512}\right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =\left(−\mathrm{512}\right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =\left(−\mathrm{8}^{\mathrm{3}} \right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =\left(−\mathrm{8}\right)^{−\mathrm{9}} \:\:\:{clear}? \\ $$
Commented by mathlove last updated on 02/Mar/23
$${tanks}\:{mr}\:{very}\:{thank} \\ $$