1.3 Murtopotenssi

167

a)
\sqrt[4]{81}=\sqrt[4]{3^4}=3
b)
\sqrt[6]{27}=\sqrt[6]{3^3}=3^{\frac{1}{2}}
c)
\sqrt[6]{9}=\sqrt[6]{3^2}=3^{\frac{2}{6}}=3^{\frac{1}{3}}
d)
\frac{\sqrt[3]{9}}{\sqrt{3}}=\frac{1}{\sqrt{3}}\cdot\sqrt[3]{3^2}=3^{-\frac{1}{2}}\cdot3^{\frac{2}{3}}=3^{-\frac{3}{6}}\cdot3^{\frac{4}{6}}=3^{\frac{1}{6}}

166

a)
\sqrt{32}=\sqrt{2^5}=2^{\frac{5}{2}}
b)
\sqrt[5]{16}=\sqrt[5]{2^4}=2^{\frac{4}{5}}
c)
\sqrt{\sqrt[3]{4}}=\sqrt{\sqrt[3]{2^2}}=2^{\frac{2}{6}}=2^{\frac{1}{3}}
d)
\frac{\sqrt[3]{16}}{\sqrt{2}}=\frac{1}{\sqrt{2}}\cdot\sqrt[3]{2^4}=2^{-\frac{1}{2}}\cdot2^{\frac{4}{3}}=2^{-\frac{3}{6}}2^{\frac{8}{6}}=2^{\frac{5}{6}}

164

a)
4^{1{,}5}=4^{\frac{3}{2}}=\sqrt{4^3}=\sqrt{4^2}\sqrt{4}=4\sqrt{4}
b)
4^{0{,}75}=4^{\frac{3}{4}}=\sqrt[4]{4^3}=\sqrt[4]{2^6}=\sqrt[4]{2^4}\sqrt[4]{2^2}=2\sqrt[4]{4}
c)
 
27^{\frac{2}{3}}=\sqrt[3]{27^2}=\sqrt[3]{3^6}=3^{\frac{6}{3}}=3^2=9
d)
25^{-\frac{3}{2}}=\frac{1}{\sqrt{25^3}}=\frac{1}{\sqrt{5^6}}=5^{-\frac{6}{2}}=5^{-3}=\frac{1}{125}

163

a)
\frac{2a^{\frac{1}{3}}}{3}=\frac{2\sqrt[3]{a}}{3}
b)
6a^{\frac{5}{2}}=6\sqrt{a^5}
c)
7a^{-\frac{4}{5}}=\frac{7}{\sqrt[5]{a^4}}
d)
\frac{a^{-\frac{9}{5}}}{4}=\frac{1}{4\sqrt[5]{a^9}}

162

162
a)
\frac{2}{\sqrt{x}}=2\cdot\ \frac{1}{\sqrt{x}}=2\sqrt{x^{-1}}=2x^{-\frac{1}{2}}
b)
\frac{x^2}{3\sqrt{x}}=\frac{1}{3}\cdot\frac{x^2}{\sqrt{x}}=\frac{1}{3}x^2\cdot x^{-\frac{1}{2}}=\frac{1}{3}x^{\frac{3}{2}}
c)
\frac{3}{4x\sqrt{x}}=\frac{3}{4}\cdot\frac{1}{x\sqrt{x}}=\frac{3}{4}x^{-1}x^{-\frac{1}{2}}=\frac{3}{2}x^{-\frac{3}{2}}
d)
\frac{2\sqrt{x}}{5x}=\frac{2}{5}\cdot\frac{\sqrt{x}}{x}=\frac{2}{5}\cdot x^{-1}\cdot x^{\frac{1}{2}}=\frac{2}{5}x^{-\frac{1}{2}}