(q^5)^gamma=q^10 ھەل قىلىش

$$( q ^ { 5 } ) ^ { \gamma } = q ^ { 10 }$$

γ نى يېشىش (complex solution)

$$\left\{\begin{matrix}\gamma =\frac{2\pi n_{1}i}{\ln(q^{5})}+10\log_{q^{5}}\left(q\right)\text{, }n_{1}\in \mathrm{Z}\text{, }&q\neq 0\text{ and }q\neq e^{\frac{8\pi i}{5}}\text{ and }q\neq e^{\frac{6\pi i}{5}}\text{ and }q\neq e^{\frac{4\pi i}{5}}\text{ and }q\neq e^{\frac{2\pi i}{5}}\text{ and }q\neq 1\\\gamma \in \mathrm{C}\text{, }&q=0\text{ or }q=e^{\frac{8\pi i}{5}}\text{ or }q=e^{\frac{6\pi i}{5}}\text{ or }q=e^{\frac{4\pi i}{5}}\text{ or }q=e^{\frac{2\pi i}{5}}\text{ or }q=1\end{matrix}\right.$$

q نى يېشىش

$$\left\{\begin{matrix}\\q=1\text{, }&\text{unconditionally}\\q=-1\text{, }&Numerator(\gamma )\text{bmod}2=0\text{ and }Denominator(\gamma )\text{bmod}2=1\\q=0\text{, }&\gamma >0\\q\neq 0\text{, }&\gamma =2\end{matrix}\right.$$

γ نى يېشىش

$$\left\{\begin{matrix}\gamma =2\text{, }&q\neq 1\text{ and }q>0\\\gamma \in \mathrm{R}\text{, }&q=1\\\gamma >0\text{, }&q=0\end{matrix}\right.$$