Pangkat Bulat

$\begin{aligned} -p \left[ -p \cdot (-p^{-3}) \right]^{-3} &= -p \left[ (-p)^{-3} \cdot (-p^{-3 \cdot -3}) \right] \\ &= -p \left[ (-p)^{-3} \cdot (-p^9) \right] \\ &= -p \left[ (-p)^{-3+9} \right] \\ &= -p \left[ (-p)^6 \right] \\ &= -p \cdot (-p)^6 \\ &= (-p)^{1+6} \\ &= (-p)^7 \\ &= -p^7 \end{aligned}$

$\begin{aligned} \left( \frac{2}{3} \right)^2 \cdot \left( -\frac{3}{4} \right)^2 - \frac{3}{2^3} : \frac{5}{8} &= \left( \frac{2}{3} \right)^2 \cdot \left( -\frac{3}{4} \right)^2 - \frac{3}{2^3} \cdot \frac{8}{5} \\ &= \frac{4}{9} \cdot \frac{9}{16} - \frac{3}{8} \cdot \frac{8}{5} \\ &= \frac{4}{16} - \frac{3}{5} \\ &= \frac{1}{4} - \frac{3}{5} \\ &= \frac{5}{20} - \frac{12}{20} \\ &= -\frac{7}{20} \\ &= -0,35 \end{aligned}$

$\begin{aligned} \frac{4^3 \cdot (3,5)^2}{4^2 \cdot 7^2} &= \frac{4^3 \cdot \left(\frac{7}{2}\right)^2}{4^2 \cdot 7^2} \\ &= \frac{4^3 \cdot \frac{7^2}{2^2}}{4^2 \cdot 7^2} \\ &= \frac{4^3 \cdot 7^2}{4^2 \cdot 7^2 \cdot 2^2} \\ &= \frac{4^3 \cdot 7^2}{4^2 \cdot 7^2 \cdot 4} \\ &= \frac{4^3 \cdot 7^2}{4^{2+1} \cdot 7^2} \\ &= \frac{4^3 \cdot 7^2}{4^3 \cdot 7^2} \\ &= 1 \end{aligned}$

$\begin{aligned} \frac{1}{1+x^{b-a}} + \frac{1}{1+x^{a-b}} &= \frac{1}{1+\frac{x^b}{x^a}} + \frac{1}{1+\frac{x^a}{x^b}} \\ &= \frac{1}{\frac{x^a}{x^a}+\frac{x^b}{x^a}} + \frac{1}{\frac{x^b}{x^b}+\frac{x^a}{x^b}} \\ &= \frac{1}{\frac{x^a + x^b}{x^a}} + \frac{1}{\frac{x^b + x^a}{x^b}} \\ &= \frac{x^a}{x^a + x^b} + \frac{x^b}{x^a + x^b} \\ &= \frac{x^a + x^b}{x^a + x^b} \\ &= 1 \end{aligned}$

$\begin{aligned} \left( \frac{8 m^{-9} \cdot n^{-2}}{64 m^{-6} \cdot n} \right)^{-1} &= \left( \frac{64 m^{-6} \cdot n}{8 m^{-9} \cdot n^{-2}} \right) \\ &= \left( \frac{64}{8} \cdot m^{-6-(-9)} \cdot n^{1-(-2)} \right) \\ &= 2^3 \cdot m^{-6+9} \cdot n^{1+2} \\ &= 2^3 \cdot m^3 \cdot n^3 \\ &= (2 \cdot m \cdot n)^3 \\ &= (2mn)^3 \end{aligned}$

$\begin{aligned} p^a \cdot (p^a)^{1-a} \cdot (p^{1+a})^a &= p^a \cdot p^{a(1-a)} \cdot p^{a(1+a)} \\ &= p^a \cdot p^{a-a^2} \cdot p^{a+a^2} \\ &= p^{a+(a-a^2)+(a+a^2)} \\ &= p^{a+a-a^2+a+a^2} \\ &= p^{3a} \end{aligned}$

$\begin{aligned} \left( \frac{a^{-4} \cdot b^2 \cdot c}{a \cdot b^{-6} \cdot c^3} \right)^4 &= \left( \frac{b^{2-(-6)}}{a^{1-(-4)} \cdot c^{3-1}} \right)^4 \\ &= \left( \frac{b^{2+6}}{a^{1+4} \cdot c^{2}} \right)^4 \\ &= \left( \frac{b^{8}}{a^{5} \cdot c^{2}} \right)^4 \\ &= \frac{b^{8 \cdot 4}}{a^{5 \cdot 4} \cdot c^{2 \cdot 4}} \\ &= \frac{b^{32}}{a^{20} \cdot c^{8}} \end{aligned}$

$\begin{aligned} 10^4 + 10^2 + 10^0 + 10^{-2} + 10^{-4} &= 10^4 + 10^2 + 10^0 + \frac{1}{10^2} + \frac{1}{10^4} \\ &= 10000 + 100 + 1 + \frac{1}{100} + \frac{1}{10000} \\ &= 10000 + 100 + 1 + 0,01 + 0,0001 \\ &= 10101,0101 \end{aligned}$

$\begin{aligned} \frac{3^{n+4} - 3 \cdot 3^{n+1}}{8 \cdot 3^{n+2}} &= \frac{3^n \cdot 3^4 - 3 \cdot 3^n \cdot 3^1}{8 \cdot 3^n \cdot 3^2} \\ &= \frac{3^n (3^4 - 3 \cdot 3)}{3^n \cdot 8 \cdot 3^2} \\ &= \frac{3^4 - 3 \cdot 3}{8 \cdot 3^2} \\ &= \frac{81 - 9}{8 \cdot 9} \\ &= \frac{72}{72} \\ &= 1 \end{aligned}$

$\begin{aligned} \frac{5^6 \cdot (6^6 - 3^8) \cdot 2^4}{11 \cdot 5^3 \cdot 10^4} &= \frac{5^6 \cdot ((2 \cdot 3)^6 - 3^{2+6}) \cdot 2^4}{11 \cdot 5^3 \cdot (2 \cdot 5)^4} \\ &= \frac{5^6 \cdot (2^6 \cdot 3^6 - 3^2 \cdot 3^6) \cdot 2^4}{11 \cdot 5^3 \cdot 2^4 \cdot 5^4} \\ &= \frac{5^6 \cdot 3^6 (2^6 - 3^2) \cdot 2^4}{11 \cdot 5^3 \cdot 2^4 \cdot 5^4} \\ &= \frac{5^6 \cdot 3^6 (2^6 - 3^2) \cdot 2^4}{11 \cdot 5^7 \cdot 2^4} \\ &= \frac{3^6 (2^6 - 3^2) \cdot 2^{4-4}}{11 \cdot 5^{7-6}} \\ &= \frac{3^6 (2^6 - 3^2) \cdot 2^0}{11 \cdot 5^1} \\ &= \frac{3^6 (64 - 9) \cdot 1}{11 \cdot 5} \\ &= \frac{3^6 (55)}{55} \\ &= 3^6 \end{aligned}$