Задание 1.3

Question

Докажите, что:
\begin{enumerate}
\item $$\frac{5^{\sqrt{8}}}{5^{\sqrt{2}}}=5^{\sqrt{2}}$$
\item $$4^{\frac{\sqrt{3}}{2}} \cdot\left(\frac{1}{8}\right)^{\sqrt{27}}=\left(16^{\sqrt{3}}\right)^{-2};$$
\item $$\frac{12^{\sqrt{48}} \cdot 2^{4 \sqrt{12}}}{4^{\sqrt{108}} \cdot 6^{\sqrt{27}}}=6^{\sqrt{3}}$$
\end{enumerate}

Gimli · Accepted Answer

\begin{enumerate}
\item $$\frac{5^{\sqrt{8}}}{5^{\sqrt{2}}}=\frac{5^{2 \sqrt{2}}}{5^{-\sqrt{2}}}=5^{2 \sqrt{2}-\sqrt{2}}=5^{\sqrt{2}}$$
\item $$\left.4^{\frac{\sqrt{3}}{2}} \cdot\left(\frac{1}{8}\right)^{\sqrt{27}}=\left(2^{2}\right)^{\frac{\sqrt{3}}{2}} \cdot\left((2^{-3}\right)^{3 \sqrt{3}}=2^{\sqrt{3}} \cdot 2^{-9 \sqrt{3}}=\left((2^{4}\right)^{\sqrt{3}}\right)^{-2}=\left(16^{\sqrt{3}}\right)^{-2}$$
\item $$\frac{12^{\sqrt{48}} \cdot 2^{4 \sqrt{12}}}{4 \sqrt{108} \cdot 6^{\sqrt{27}}}=\frac{(2 \cdot 6)^{2 \sqrt{12}} \cdot 2^{4 \sqrt{12}}}{\left(2^{2}\right)^{3 \sqrt{12}} \cdot 6^{3 \sqrt{3}}}=\frac{2^{2 \sqrt{12}+4 \sqrt{12}} \cdot 6^{2 \sqrt{12}}}{2^{6 \sqrt{2}} \cdot 6^{3 \sqrt{2}}}=\frac{2^{6 \sqrt{2}} \cdot 6^{4 \sqrt{3}}}{2^{6 \sqrt{12}} \cdot 6^{3 \sqrt{3}}}=$$
$$=6^{4 \sqrt{3}-3 \sqrt{3}}=6^{\sqrt{3}} .$$\
\end{enumerate}

Задание 1.3

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Задание 1.3

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