Oppgåve

Potensar og rotuttrykk

Oppgåvene skal løysast utan bruk av hjelpemiddel med mindre det står noko anna. Nedst på sida kan du laste ned oppgåvene som Word- og pdf-dokument.

Oppgåve 1

Rekn ut.

a) $\sqrt[3]{8}$

Løysing

$\sqrt[3]{8} = 2$ fordi $2^{3} = 8$

eller

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

b) $\sqrt[3]{27}$

Løysing

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

eller

$\sqrt[3]{27} = 27^{\frac{1}{3}} = {(3^{3})}^{\frac{1}{3}} = 3^{3 \cdot \frac{1}{3}} = 3^{1} = 3$

c) $\sqrt[4]{81}$

Løysing

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

$\sqrt[4]{81} = 81^{\frac{1}{4}} = {(3^{4})}^{\frac{1}{4}} = 3^{4 \frac{1}{4}} = 3^{1} = 3$

Oppgåve 2

Rekn ut med hjelpemiddel.

a) $\sqrt[5]{15, 25}$

Løysing

$\sqrt[5]{15, 25} = 1, 72$

b) $\sqrt[8]{100}$

Løysing

$\sqrt[8]{100} = 1, 78$

c) $\sqrt[9]{2, 25}$

Løysing

$\sqrt[9]{2, 25} = 1, 09$

Oppgåve 3

Rekn ut.

a) $9^{\frac{1}{2}}$

Løysing

$9^{\frac{1}{2}} = \sqrt{9} = 3$

b) $27^{\frac{1}{3}}$

Løysing

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

c) $\sqrt[5]{32}$

Løysing

$\sqrt[5]{32} = 2$

d) $256^{\frac{1}{4}}$

Løysing

$256^{\frac{1}{4}} = \sqrt[4]{256} = 4$

Oppgåve 4

Rekn ut.

a) $\sqrt[3]{125}$

Løysing

$\sqrt[3]{125} = 5$

b) $\sqrt[3]{9} \cdot \sqrt[3]{3}$

Løysing

$\sqrt[3]{9} \cdot \sqrt[3]{3} = \sqrt[3]{9 \cdot 3} = \sqrt[3]{27} = 3$

c) $\frac{\sqrt[3]{54}}{\sqrt[3]{2}}$

Løysing

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

Oppgåve 5

Vis at

a) $8^{\frac{2}{3}} = 4$

Løysing

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

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

Løysing

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

Oppgåve 6

Vis at

a) $4^{\frac{3}{2}} = 8$

Løysing

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

b) $9^{\frac{3}{2}} = 27$

Løysing

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

c) $3^{\frac{4}{3}} = 3 \cdot \sqrt[3]{3}$

Løysing

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

d) $5^{\frac{5}{3}} = 5 \cdot \sqrt[3]{25}$

Løysing

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

e) $2^{\frac{7}{3}} = 4 \cdot \sqrt[3]{2}$

Løysing

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

Oppgåve 7

Rekn ut.

a) $27^{- \frac{1}{3}}$

Løysing

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

b) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}}$

Løysing

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

c) $4^{\frac{5}{2}} \cdot 4^{- \frac{3}{2}}$

Løysing

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

d) $2^{\frac{5}{2}} \cdot 4^{- \frac{3}{2}}$

Løysing

$\begin{array}{rcl} 2^{\frac{5}{2}} \cdot 4^{- \frac{3}{2}} & = & 2^{\frac{5}{2}} \cdot {(2^{2})}^{- \frac{3}{2}} = 2^{\frac{5}{2} - \frac{6}{2}} = 2^{- \frac{1}{2}} \\ = & \frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2} \end{array}$

e) ${(3^{4})}^{\frac{1}{4}}$

Løysing

${(3^{4})}^{\frac{1}{4}} = 3^{4 \cdot \frac{1}{4}} = 3^{\frac{4}{4}} = 3$

f) $\frac{3^{\frac{4}{3}} \cdot 3^{\frac{1}{3}}}{3^{- \frac{1}{3}}}$

Løysing

$\begin{array}{rcl} \frac{3^{\frac{4}{3}} \cdot 3^{\frac{1}{3}}}{3^{- \frac{1}{3}}} & = & 3^{\frac{4}{3} + \frac{1}{3} - (- \frac{1}{3})} = 3^{\frac{4}{3} + \frac{1}{3} + \frac{1}{3}} \\ = & 3^{\frac{6}{3}} = 3^{2} = 9 \end{array}$

g) $\frac{4^{\frac{4}{3}} \cdot 2^{\frac{1}{3}}}{2^{2}}$

Løysing

$\frac{4^{\frac{4}{3}} \cdot 2^{\frac{1}{3}}}{2^{2}} = \frac{2^{\frac{8}{3}} \cdot 2^{\frac{1}{3}}}{2^{2}} = 2^{\frac{8}{3} + \frac{1}{3} - 2} = 2^{1} = 2$