Ännu mer problem med potensregler

Uppgift 1:

Täljare:

$$\begin{gathered} (8+7i)(6-7i)-(8\times 6)+8\times (-7i)+7i\times 6+7i\times (-7i)\hspace{0.17em}=\hspace{0.17em} \\ =48-56i+42i-49i^2= \\ =48-14i-49(-1) \\ =97-14i \end{gathered}$$

Nämnare:

$$\begin{gathered} (6+7i)(6-7i)-(6\times 6)+(6\times (-7i\left)\right)+7i\times 6+7i\times (-7i) = \\ 36-42i+42i-49i^2= \\ 36-49(-1)= \\ 36+49=85 \end{gathered}$$

Täljare/Nämnare:

$$\begin{gathered} \frac{97-14i}{85}= z \\ z=\frac{97}{85}-\frac{14}{85}i \end{gathered}$$

Uppgift 2:

$$\begin{gathered} (a+bi)\times (c+di)-(ac-bd)+(ad+bc)i \\ a=3 \\ b=-6 \\ c=4 \\ d=2 \\ (3-6i)\times (4+2i)-\left(\right(3\times 4)-((-6)\times 2\left)\right)+\left(\right(3\times 2)+(4\times (-6)\left)\right)i \\ =(12+12)+(6-24)i \\ =24+18i=\hspace{0.17em}z \end{gathered}$$

Uppgift 3:

Täljare:

$$\begin{gathered} 5^4\times \left({25}^3\right)\times 5^{1/4} \\ = 5^4\times {\left(5^2\right)}^3\times 5^{1/4} \\ =5^4\times 5^6\times 5^{1/4} \\ =5^{4+6+(1/4)} \\ =5^{10+(1/4)} \end{gathered}$$

Nämnare:

$$\begin{gathered} \sqrt[2]{5}\times 5^3\times {125}^5 \\ =5^{1/2}\times 5^3\times {\left(5^3\right)}^5 \\ =5^{1/2}\times 5^3\times 5^{15} \\ =5^{1/2}+5^3+5^{15} \\ =5^{(1/2)+3+15} \\ =5^{(31/2)} \end{gathered}$$

Täljare/Nämnare:

$$\begin{gathered} \frac{5^{10+(1/4)}}{5^{(31/2)}} \\ =5^{(10+(1/4)-(31/2)} \\ =5^{(10+(1/4\left)\right)-(15\times (1/2\left)\right)} \\ =5^{10-(31/2)+(1/4)} \end{gathered}$$

Skriva om: $\frac{31}{2}=\frac{62}{4}$

$$\begin{gathered} 5^{10-(62/4)+(1/4)} \\ =5^{10-(61/4)} \\ =5^{10-15.25} \\ =5^{-5.25} \end{gathered}$$