توان

$\begin{array}{l}={\left(\frac{6}{2}\right)}^3\\ \\ =3^3\end{array}$

$\begin{array}{l}={\left(\frac{2}{{\displaystyle \frac{1}{3}}}\right)}^7\\ \\ ={\left(\frac{{\displaystyle \frac{2}{1}}}{{\displaystyle \frac{1}{3}}}\right)}^7\\ \\ ={\left(\frac{6}{1}\right)}^7\\ \\ =6^7\end{array}$

$\begin{array}{l}=\frac{2^4\times a^4}{3^4}\\ \\ =\frac{{\left(2\times a\right)}^4}{3^4}\\ \\ =\frac{{\left(2a\right)}^4}{3^4}\\ \\ ={\left(\frac{2a}{3}\right)}^4\end{array}$

$\begin{array}{l}=\frac{a^{5+2}}{b^{12-5}}\\ \\ =\frac{a^7}{b^7}\\ \\ ={\left(\frac{a}{b}\right)}^7\end{array}$

$\begin{array}{l}=\frac{a^{5+2+1}\times b^{3+4+1}}{{\left(a\times b\right)}^7}\\ \\ =\frac{a^8\times b^8}{{\left(a\times b\right)}^7}\\ \\ =\frac{{\left(a\times b\right)}^8}{{\left(a\times b\right)}^7}\\ \end{array}$

$\begin{array}{l}={\left(a\times b\right)}^8\times {\left(a\times b\right)}^{-7}\\ \\ ={\left(a\times b\right)}^{8-7}\\ \\ ={\left(a\times b\right)}^1\end{array}$

$\begin{array}{l}=\frac{a^7}{a^2\times b^5}\\ \\ =\frac{a^2\times a^5}{a^2\times b^5}\\ \\ =\frac{a^5}{b^5}\\ \\ ={\left(\frac{a}{b}\right)}^5\end{array}$

$\begin{array}{l}=\frac{a^{7-4}}{b^{5-2}}\\ \\ =\frac{a^3}{b^3}\\ \\ ={\left(\frac{a}{b}\right)}^3\end{array}$

$\begin{array}{l}=\frac{\left(a^4\times a^3\right)\times \left(b^4\times b\right)}{\left(a^2\times a\right)\times \left(b^7\times b^2\right)}\\ \\ =\frac{a^7\times b^5}{a^3\times b^9}\\ \\ =\frac{a^3\times a^4\times b^5}{a^3\times b^5\times b^4}\\ \end{array}$

$\begin{array}{l}=\frac{a^4}{b^4}\\ \\ ={\left(\frac{a}{b}\right)}^4\end{array}$

$\begin{array}{l}={\left(\frac{12}{ \frac{2}{3}}\right)}^9÷6^9\\ \\ ={\left(\frac{3\times 12}{2}\right)}^9÷6^9\\ \\ ={18}^9÷6^9\\ \end{array}$

$\begin{array}{l}={\left(\frac{18}{6}\right)}^9\\ \\ =3^9\end{array}$

$\begin{array}{l}=\frac{a^{7+3}\times b^{4+6}}{{\left(a\times b\right)}^{10}}\\ \\ =\frac{a^{10}\times b^{10}}{{\left(a\times b\right)}^{10}}\\ \\ =\frac{{\left(a\times b\right)}^{10}}{{\left(a\times b\right)}^{10}}\\ \\ =1\end{array}$

$\begin{array}{l}=\frac{a^{1375-74}\times a^{99}}{b^{200+500+700}}\\ \\ =\frac{a^{1301}\times a^{99}}{b^{1400}}\\ \end{array}$

$\begin{array}{l}=\frac{a^{1400}}{b^{1400}}\\ \\ ={\left(\frac{a}{b}\right)}^{1400}\end{array}$

$\begin{array}{l}=\frac{ \frac{{1440}^{15}}{{12}^{22}}}{ \frac{{15}^{30}}{6^{37}}}\\ \\ =\frac{{1440}^{15}\times 6^{37}}{{15}^{30}\times {12}^{22}}\\ \\ =\frac{{\left(12\times 120\right)}^{15}\times 6^{37}}{{15}^{30}\times {12}^{22}}\\ \end{array}$

$\begin{array}{l}=\frac{{12}^{15}\times {120}^{15}\times 6^{37}}{{15}^{30}\times {12}^{22}}\\ \\ =\frac{{120}^{15}\times 6^{37}}{{15}^{30}\times {12}^7}\\ \end{array}$

$\begin{array}{l}=\frac{{\left(15\times 8\right)}^{15}\times 6^{37}}{{15}^{30}\times {12}^7}\\ \\ =\frac{{15}^{15}\times 8^{15}\times 6^{37}}{{15}^{30}\times {12}^7}\\ \end{array}$

$\begin{array}{l}=\frac{8^{15}\times 6^{37}}{{15}^{15}\times {12}^7}\\ \\ =\frac{{\left(2\times 4\right)}^{15}\times 6^{37}}{{\left(15\right)}^{15}\times {\left(2\times 6\right)}^7}\\ \end{array}$

$\begin{array}{l}=\frac{2^{15}\times 4^{15}\times 6^{37}}{{15}^{15}\times 2^7\times 6^7}\\ \\ =\frac{2^8\times 4^{15}\times 6^{30}}{{15}^{15}}\\ \end{array}$

$\begin{array}{l}=\frac{2^8\times 2^{30}\times {\left(3\times 2\right)}^{30}}{{\left(3\times 5\right)}^{15}}\\ \\ =\frac{2^8\times 2^{30}\times 3^{30}\times 2^{30}}{3^{15}\times 5^{15}}\\ \\ =\frac{2^8\times 2^{30}\times 2^{30}\times 3^{15}}{5^{15}}\\ \end{array}$

$\begin{array}{l}=\frac{2^{68}\times 3^{15}}{5^{15}}\\ =2^{68}\times {\left(\frac{3}{5}\right)}^{15}\end{array}$