Power formulas are based on the same observations as you made in task 1 and 2. All formulas are written below. If you have a formula book in use, please, check how they are written there.[br] [br]  [math]\Large\begin{array}{lrcll}[br]\textcolor{blue}{1:}&\textcolor{blue}{ a^ma^n}&\textcolor{blue}{=}&\textcolor{blue}{a^{m+n}:}& 5^2\cdot 5^3=5^{2+3}=5^5=\overbrace{5 \cdot 5}^{2}\cdot \overbrace{5 \cdot 5\cdot 5}^{3}\\ \vspace{2mm}\\[br]\textcolor{blue}{2:}&\textcolor{blue}{ \frac{a^m}{a^n}}&\textcolor{blue}{=}&\textcolor{blue}{a^{m-n}:}&\frac{5^6}{5^2}=5^{6-2}=5^4=\frac{\cancel 5\cdot \cancel 5 \cdot 5 \cdot 5\cdot 5\cdot 5}{\cancel 5\cdot \cancel 5}\\\vspace{2mm}\\[br]\textcolor{blue}{3:}& \textcolor{blue}{(ab)^n}&\textcolor{blue}{=}&\textcolor{blue}{a^nb^n:}&(3\cdot 4)^2=3\cdot 4 \;\cdot \;3\cdot 4 = 3\cdot 3 \cdot 4 \cdot 4 = 3^2\cdot 4^2\\\vspace{2mm}\\[br]\textcolor{blue}{4:}& \textcolor{blue}{\left (\frac{a}{b}\right )^n}&\textcolor{blue}{=}&\textcolor{blue}{\frac{a^n}{b^n}:}&\left (\frac{3}{4}\right ) ^2= \frac{3}{4}\cdot \frac{3}{4}=\frac{3\cdot 3}{4\cdot 4}=\frac{3^2}{4^2}\\\vspace{2mm}\\[br]\textcolor{blue}{5:}& \textcolor{blue}{\left ( a^m\right )^n}&\textcolor{blue}{=}&\textcolor{blue}{a^{mn}:} &\left ( 5^3\right )^2 = 5^3\cdot 5^3 = 5^{3+3} = 5^{2\cdot 3}=5^6[br]\end{array}[br][/math][br] [br][br]Based on these formulas: [math]\Large \textcolor{blue}{a^0=1}.[/math] We can check this with numbers [math]\frac{a^2}{a^2}=a^{2-2}=a^0 =\frac{\cancel a\cdot \cancel a}{\cancel a\cdot \cancel a}=1.[/math][br][br][color=#0000ff]Example 1.[/color][br][br][math] \frac{2^3\cdot 3^3}{36}=\frac{2^3\cdot 3^3}{4\cdot 9}=\frac{2^3\cdot 3^3}{2^2\cdot 3^2}=2^{3-2}\cdot 3^{3-2}=2\cdot 3=6[/math][br] [br]and the same without the actual numbers:[br][br]  [math] \frac{a^3\cdot b^3}{a^2\cdot b^2}=a^{3-2}\cdot b^{3-2}=a\cdot b[/math][br] [br] [br][color=#0000ff]Example 2[/color].[br][br]  [math] \left ( \frac{8\cdot 27}{72}\right )^2 =\left (\frac{8\cdot 3^3}{8\cdot \underbrace{9}_{3^2}} \right )^2=(3^{3-2})^2=3^2=9[/math][br] [br][br][color=#0000ff]Example 3[/color]. [br][br]  [math] \left (\frac{x^2\cdot y^3}{z^4}\right )^2=\frac{(x^2)^2\cdot (y^3)^2}{(z^4)^2}==\frac{x^{2\cdot 2}\cdot y^{3\cdot 2}}{z^{4\cdot 2}}=\frac{x^4y^6}{z^8}.[/math][br] [br][br][color=#0000ff]Example 4.[/color][br][br] [br]  [math]\frac{a^6b^3}{a^4b^5}=\frac{a^6}{a^4}\cdot\frac{b^3}{b^5}=a^{6-4}\cdot b^{3-5}=a^2\cdot b^{-2}=\frac{a^2}{b^2}=\left (\frac{a}{b}\right )^2.[/math][br]