Potenser og rotuttrykk

$$\sqrt[3]{8}=2$$ fordi $$2^3=8$$. Alternativ løsningsmåte:

$$\sqrt[3]{8}=8^{\frac{1}{3}}={\left(2^3\right)}^{\frac{1}{3}}=2^{3·\frac{1}{3}}=2^1=2$$

$$\sqrt[3]{27}=3$$ fordi $$3^3=27$$. Alternativ løsningsmåte:

$$\sqrt[3]{27}={27}^{\frac{1}{3}}={\left(3^3\right)}^{\frac{1}{3}}=3^{3·\frac{1}{3}}=3^1=3$$

$$\sqrt[4]{81}=3$$ fordi $$3^4=81$$

$$={81}^{\frac{1}{4}}={\left(3^4\right)}^{\frac{1}{4}}=3^{4\frac{1}{4}}=3^1=3$$

$$=1,72$$

$$=1,78$$

$$=1,09$$

$$=\sqrt{9}=3$$

$$=\sqrt[3]{27}=3$$

$$=2$$

$$=\sqrt[4]{256}=4$$

$$=5$$

$$=\sqrt[3]{9·3}=\sqrt[3]{27}=3$$

$$\sqrt[3]{\frac{54}{2}}=\sqrt[3]{27}=3$$

$$\begin{array}{c}\begin{array}{l}{8}^{\frac{2}{3}}={\left(2^3\right)}^{\frac{2}{3}}=2^{3\frac{2}{3}}=2^2=4\\ \\ 8^{\frac{2}{3}}=\sqrt[3]{8^2}=\sqrt[3]{64}=4\end{array}\end{array}$$

$$6^{\frac{2}{3}}=\sqrt[3]{6^2}=\sqrt[3]{36}$$

$$4^{\frac{3}{2}}=\sqrt{4^3}=4\sqrt{4}=4·2=8$$

$$9^{\frac{3}{2}}=\sqrt{9^3}=9\sqrt{9}=9·3=27$$

$$3^{\frac{4}{3}}=3^{1+\frac{1}{3}}=3^1·3^{\frac{1}{3}}=3·\sqrt[3]{3}$$

$$5^{\frac{3}{5}}=5^{1+\frac{1}{3}}=5^1·5^{\frac{2}{3}}=5\sqrt[3]{5^2}=5·\sqrt[3]{25}$$

$$2^{\frac{7}{3}}=2^{2+\frac{1}{3}}=2^2·2^{\frac{1}{3}}=4·\sqrt[3]{2}$$

$$=\frac{1}{{27}^{{\displaystyle \frac{1}{3}}}}=\frac{1}{\sqrt[3]{27}}=\frac{1}{3}$$

$$=2^{\frac{2}{3}+\frac{1}{3}}=2^{\frac{3}{3}}=2$$

$$=4^{\frac{5}{2}-\frac{3}{2}}=4^{\frac{2}{2}}=4$$

$$=2^{\frac{5}{2}}·{\left(2^2\right)}^{-\frac{3}{2}}=2^{\frac{5}{2}-\frac{6}{2}}=2^{-\frac{1}{2}}=\frac{1}{\sqrt{2}}=\frac{1·\sqrt{2}}{\sqrt{2}·\sqrt{2}}=\frac{\sqrt{2}}{2}$$

$$=3^{4\frac{1}{4}}=3^{\frac{4}{4}}=3$$

$$=3^{\frac{4}{3}+\frac{1}{3}-\left(-\frac{1}{3}\right)}=3^{\frac{4}{3}+\frac{1}{3}+\frac{1}{3}}=3^{\frac{6}{3}}=3^2=9$$

$$=\frac{2^{{\displaystyle \frac{8}{3}}}·2^{{\displaystyle \frac{1}{3}}}}{2^2}=2^{\frac{8}{3}+\frac{1}{3}-2}=2^1=2$$

Vi må løse likningen $$\begin{array}{rcl}4\pi r^2& = & 17\end{array}$$.

Radien er 1,16 cm.

Vi må løse likningen $$\begin{array}{rcl}\frac{4\pi r^3}{3}& = & 9,35\end{array}$$.

Radien er 1,31 cm.