\(\dfrac{(m^{\frac{1}{2}}\cdot n^{-2})^4\:(m^{-\frac{12}{5}}\cdot n^4)^5}{(m^{\frac{5}{2}}\cdot n^8)^2}\)

\(\dfrac{(m^{\frac{1}{2}(4)}\cdot n^{-2(4)})\:(m^{-\frac{12}{5}(5)}\cdot n^{4(5)})}{(m^{\frac{5}{2}(2)}\cdot n^{8(2)})}\)

\(\dfrac{m^2\cdot n^{-8}\cdot m^{-12}\cdot n^{20}}{m^5\cdot n^{16}}\)

\(\dfrac{m^{2 + (-12)}\cdot n^{-8 + 20}}{m^5\cdot n^{16}}\)

\(m^{-10-5}\cdot n^{12-16}\)

\(m^{-15}\cdot n^{-4}\)

\(\dfrac{1}{m^{15}\:n^4}\)

Buatlah faktorisasi prima untuk masing-masing angka

\(\dfrac{(2^2\cdot 3 \cdot 5)^7}{(3\cdot 5)^7}\div \dfrac{(2\cdot 3^4)^5}{(2^3\cdot 5)^5}\times \dfrac{(2^5)^3}{(2\cdot 3\cdot 5^2)^3}\)

\(\dfrac{2^{14}\cdot \cancel{3^7\cdot 5^7}}{\cancel{3^7\cdot 5^7}}\div \dfrac{2^5\cdot 3^{20}}{2^{15}\cdot 5^5}\times \dfrac{2^{15}}{2^3\cdot 3^3\cdot 5^6}\)

\(2^{14}\times \dfrac{2^{15}\cdot 5^5}{2^5\cdot 3^{20}}\times \dfrac{2^{15-3}}{3^3\cdot 5^6}\)

\(2^{14}\times 2^{15-5} \cdot 3^{-20} \cdot 5^5 \times 2^{12}\cdot 3^{-3}\cdot 5^{-6}\)

\(2^{14}\cdot 2^{10}\cdot 2^{12}\cdot 3^{-20}\cdot 3^{-3}\cdot 5^5\cdot 5^{-6}\)

\( 2^{14+10+12}\cdot 3^{-20-3}\cdot 5^{5-6}\)

\(2^{36}\cdot 3^{-23}\cdot 5^{-1}\)

\(\dfrac{2^{36}}{3^{23}\cdot 5}\)

\(\left(\dfrac{8(x\:-\: y)(xy)^{-1}}{4xy^{-2} – 4yx^{-2}}\right)^{-1}\)

\(\color{cyan} \text{Note: }\:\left(\dfrac{a}{b}\right)^{-1} = \left(\dfrac{b}{a}\right)^{1}\)

\(\dfrac{4xy^{-2}\:-\: 4yx^{-2}}{8(x\:-\: y)(xy)^{-1}}\)

\(\dfrac{\frac{4x}{y^2}\:-\: \frac{4y}{x^2}}{\frac{8(x\:-\: y)}{xy}}\)

\(\dfrac{\frac{4x^3\:-\: 4y^3}{(xy)^2}}{\frac{8(x\:-\: y)}{xy}}\)

\(\dfrac{4(x^3\:-\: y^3)}{\cancelto{xy}{(xy)^2}}\times \dfrac{\cancel{xy}}{8(x\:-\: y)}\)

\(\color{cyan} \text{Note: }\:x^3 \:-\: y^3 = (x\:-\: y)(x^2 + xy + y^2)\)

\(\dfrac{4\cancel{(x\:-\: y)}(x^2+xy+y^2)}{xy\cdot 8\cancel{(x\:-\: y)}}\)

\(\dfrac{x^2+xy+y^2}{2xy}\)

\(\dfrac{(2^5\cdot 2^{n+1} \:-\: 2^{-2}\cdot 2^{n + 3})}{2}\times 2^{-n\:-\: 3}\)

\(\color{cyan}\text{Note: }\:a^{m + n} = a^m\cdot a^n\)

\((\dfrac{2^5\cdot 2^n\cdot 2^1 \:-\: 2^{-2}\cdot 2^n\cdot 2^3}{2})\times 2^{-n}\cdot 2^{-3}\)

\((\dfrac{2^n\cdot 2^{5+1} \:-\: 2^n\cdot 2^{-2+3}}{2})\times \dfrac{1}{2^n\cdot 2^3}\)

\(\dfrac{\cancel{2^n}\cdot (2^6\:-\: 2)}{\cancel{2^n}\cdot 2^{1+3}}\)

\(\dfrac{64\:-\: 2}{16}\)

\(\dfrac{62}{16}\)

\(\dfrac{31}{8}\)

\(\dfrac{2^{m+2}\cdot 14^{m\:-\: 1}\cdot 7^{m+1}}{196^m}\)

\(\dfrac{2^m\cdot 2^2\cdot 14^m\cdot 14^{-1}\cdot 7^m\cdot 7^1}{14^{2m}}\)

\(\dfrac{2^m\cdot 7^m\cdot 14^m\cdot 2^2\cdot \frac{1}{14}\cdot 7}{14^{2m}}\)

\(\dfrac{(2\cdot 7)^m\cdot 14^m\cdot 2}{14^{2m}}\)

\(\dfrac{14^m\cdot 14^m\cdot 2}{14^{2m}}\)

\(\dfrac{\cancel{14^{2m}}\cdot 2}{\cancel{14^{2m}}}\)

\(2\)