# Oppgaver: Tallregning :::::::::::::::{exercise} Oppgave 1 --- level: 1 --- Regn ut. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 2 \cdot 3 + 5 - 2\cdot 6 $$ ::::{answer} $-1$ :::: ::::{solution} \begin{align*} 2 \cdot 3 + 5 - 2\cdot 6 &= 6 + 5 - 12\\ \\ &= 11 - 12\\ \\ &= -1 \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} b $$ 4 \cdot 2 - 7 + 3 \cdot 5 $$ ::::{answer} $16$ :::: ::::{solution} \begin{align*} 4 \cdot 2 - 7 + 3 \cdot 5 &= 8 - 7 + 15\\ \\ &= 1 + 15\\ \\ &= 16 \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} c $$ 8 - 3 \cdot 2 + 6 \cdot 2 $$ ::::{answer} $14$ :::: ::::{solution} \begin{align*} 8 - 3 \cdot 2 + 6 \cdot 2 &= 8 - 6 + 12\\ \\ &= 2 + 12\\ \\ &= 14 \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} d $$ 5 \cdot 4 - 2 \cdot 7 + 1 $$ ::::{answer} $7$ :::: ::::{solution} \begin{align*} 5 \cdot 4 - 2 \cdot 7 + 1 &= 20 - 14 + 1\\ \\ &= 6 + 1\\ \\ &= 7 \end{align*} :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 2 --- level: 1 --- Primtallsfaktoriser tallene. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 12 $$ ::::{answer} $$ 12 = 2^2 \cdot 3 $$ :::: ::::{solution} :::{factor-tree} n: 12 align: right width: 60% ::: $$ 12 = 4 \cdot 3 = 2^2 \cdot 3 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ 30 $$ ::::{answer} $$ 30 = 2 \cdot 3 \cdot 5 $$ :::: ::::{solution} :::{factor-tree} n: 30 align: right width: 60% ::: $$ 30 = 2 \cdot 3 \cdot 5 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ 62 $$ ::::{answer} $$ 62 = 2 \cdot 31 $$ :::: ::::{solution} :::{factor-tree} n: 62 align: right width: 60% ::: $$ 62 = 2 \cdot 31 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ 102 $$ ::::{answer} $$ 102 = 2 \cdot 3 \cdot 17 $$ :::: ::::{solution} :::{factor-tree} n: 102 align: right width: 60% ::: $$ 102 = 2 \cdot 3 \cdot 17 $$ :::: :::::::::::: :::::::::::::: ::::::::::::::: --- ::::::::::::{exercise} Oppgave 3 --- level: 1 --- Forkort brøkene så mye som mulig. :::::::::::{tab-set} --- class: tabs-parts --- ::::::::::{tab-item} a $$ \dfrac{4}{8} $$ ::::{answer} $$ \dfrac{1}{2} $$ :::: ::::{solution} $$ \dfrac{4}{8} = \dfrac{1\cdot 4}{2\cdot 4} = \dfrac{1}{2} $$ :::: :::::::::: ::::::::::{tab-item} b $$ \dfrac{6}{9} $$ ::::{answer} $$ \dfrac{2}{3} $$ :::: ::::{solution} $$ \dfrac{6}{9} = \dfrac{2\cdot 3}{3\cdot 3} = \dfrac{2}{3} $$ :::: :::::::::: ::::::::::{tab-item} c $$ \dfrac{18}{40} $$ ::::{answer} $$ \dfrac{9}{20} $$ :::: ::::{solution} $$ \dfrac{18}{40} = \dfrac{2\cdot 9}{2\cdot 20} = \dfrac{9}{20} $$ :::: :::::::::: ::::::::::{tab-item} d $$ \dfrac{9}{36} $$ ::::{answer} $$ \dfrac{1}{4} $$ :::: ::::{solution} $$ \dfrac{9}{36} = \dfrac{1\cdot 9}{4\cdot 9} = \dfrac{1}{4} $$ :::: :::: :::::::::: ::::::::::: :::::::::::: --- ::::::::::::{exercise} Oppgave 4 --- level: 1 --- Regn ut og forkort svaret så mye som mulig. :::::::::::{tab-set} --- class: tabs-parts --- ::::::::::{tab-item} a $$ \dfrac{1}{4} + \dfrac{2}{3} $$ ::::{answer} $$ \dfrac{11}{12} $$ :::: ::::{solution} $$ \dfrac{1}{4} + \dfrac{2}{3} = \dfrac{1\cdot 3}{4\cdot 3} + \dfrac{2 \cdot 4}{3 \cdot 4} = \dfrac{3}{12} + \dfrac{8}{12} = \dfrac{11}{12} $$ :::: :::::::::: ::::::::::{tab-item} b $$ \dfrac{1}{3} - \dfrac{7}{12} $$ ::::{answer} $$ -\dfrac{1}{4} $$ :::: ::::{solution} $$ \dfrac{1}{3} - \dfrac{7}{12} = \dfrac{1\cdot 4}{3\cdot 4} - \dfrac{7}{12} = \dfrac{4}{12} - \dfrac{7}{12} = -\dfrac{3}{12} = -\dfrac{1}{4} $$ :::: :::::::::: ::::::::::{tab-item} c $$ \dfrac{2}{5} + \dfrac{3}{10} $$ ::::{answer} $$ \dfrac{7}{10} $$ :::: ::::{solution} $$ \dfrac{2}{5} + \dfrac{3}{10} = \dfrac{2\cdot 2}{5\cdot 2} + \dfrac{3}{10} = \dfrac{4}{10} + \dfrac{3}{10} = \dfrac{7}{10} $$ :::: :::::::::: ::::::::::{tab-item} d $$ \dfrac{5}{6} - \dfrac{1}{4} $$ ::::{answer} $$ \dfrac{7}{12} $$ :::: ::::{solution} $$ \dfrac{5}{6} - \dfrac{1}{4} = \dfrac{5\cdot 2}{6\cdot 2} - \dfrac{1 \cdot 3}{4 \cdot 3} = \dfrac{10}{12} - \dfrac{3}{12} = \dfrac{7}{12} $$ :::: :::::::::: ::::::::::: :::::::::::: --- :::::::::::::::{exercise} Oppgave 5 --- level: 2 --- Regn ut. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 2 \cdot \dfrac{3}{4} + 5 - 4\cdot \dfrac{1}{2} $$ ::::{answer} $$ \dfrac{9}{2} $$ :::: ::::{solution} \begin{align*} 2 \cdot \dfrac{3}{4} + 5 - 4\cdot \dfrac{1}{2} &= \dfrac{6}{4} + 5 - \dfrac{4}{2}\\ \\ &= \dfrac{3}{2} + 5 - 2\\ \\ &= \dfrac{3}{2} + 3\\ \\ &= \dfrac{3}{2} + \dfrac{6}{2}\\ \\ &= \dfrac{9}{2} \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} b $$ 3 \cdot \dfrac{5}{6} - 2 + \dfrac{1}{3} $$ ::::{answer} $$ \dfrac{5}{6} $$ :::: ::::{solution} \begin{align*} 3 \cdot \dfrac{5}{6} - 2 + \dfrac{1}{3} &= \dfrac{15}{6} - 2 + \dfrac{1}{3}\\ \\ &= \dfrac{5}{2} - 2 + \dfrac{1}{3}\\ \\ &= \dfrac{15}{6} - \dfrac{12}{6} + \dfrac{2}{6}\\ \\ &= \dfrac{15 - 12 + 2}{6}\\ \\ &= \dfrac{5}{6} \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{7}{8} + 2 \cdot \dfrac{1}{4} - \dfrac{3}{2} $$ ::::{answer} $$ -\dfrac{1}{8} $$ :::: ::::{solution} \begin{align*} \dfrac{7}{8} + 2 \cdot \dfrac{1}{4} - \dfrac{3}{2} &= \dfrac{7}{8} + \dfrac{2}{4} - \dfrac{3}{2}\\ \\ &= \dfrac{7}{8} + \dfrac{1}{2} - \dfrac{3}{2}\\ \\ &= \dfrac{7}{8} + \dfrac{4}{8} - \dfrac{12}{8}\\ \\ &= \dfrac{7 + 4 - 12}{8}\\ \\ &= \dfrac{-1}{8}\\ \\ &= -\dfrac{1}{8} \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} d $$ 5 - \dfrac{2}{3} \cdot 4 + \dfrac{1}{6} $$ ::::{answer} $$ \dfrac{5}{2} $$ :::: ::::{solution} \begin{align*} 5 - \dfrac{2}{3} \cdot 4 + \dfrac{1}{6} &= 5 - \dfrac{8}{3} + \dfrac{1}{6}\\ \\ &= \dfrac{30}{6} - \dfrac{16}{6} + \dfrac{1}{6}\\ \\ &= \dfrac{30 - 16 + 1}{6}\\ \\ &= \dfrac{15}{6}\\ \\ &= \dfrac{5}{2} \end{align*} :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- ::::::::::::{exercise} Oppgave 6 --- level: 2 --- Regn ut og skriv svaret så enkelt som mulig. :::::::::::{tab-set} --- class: tabs-parts --- ::::::::::{tab-item} a $$ 2 \cdot \dfrac{3}{4} + 5 \cdot \dfrac{1}{8} $$ ::::{answer} $$ \dfrac{17}{8} $$ :::: ::::{solution} \begin{align*} 2 \cdot \dfrac{3}{4} + 5 \cdot \dfrac{1}{8} &= \dfrac{6}{4} + \dfrac{5}{8}\\ \\ &= \dfrac{3}{2} + \dfrac{5}{8}\\ \\ &= \dfrac{12}{8} + \dfrac{5}{8}\\ \\ &= \dfrac{17}{8} \end{align*} :::: :::::::::: ::::::::::{tab-item} b $$ \dfrac{1}{2} \cdot 3 + \dfrac{1}{4} \cdot 8 $$ ::::{answer} $$ \dfrac{7}{2} $$ :::: ::::{solution} \begin{align*} \dfrac{1}{2} \cdot 3 + \dfrac{1}{4} \cdot 8 &= \dfrac{3}{2} + \dfrac{8}{4}\\ \\ &= \dfrac{3}{2} + 2\\ \\ &= \dfrac{3}{2} + \dfrac{4}{2}\\ \\ &= \dfrac{7}{2} \end{align*} :::: :::::::::: ::::::::::{tab-item} c $$ \dfrac{2}{3} \cdot \dfrac{3}{4} + \dfrac{1}{2} \cdot \dfrac{5}{6} $$ ::::{answer} $$ \dfrac{11}{12} $$ :::: ::::{solution} \begin{align*} \dfrac{2}{3} \cdot \dfrac{3}{4} + \dfrac{1}{2} \cdot \dfrac{5}{6} &= \dfrac{6}{12} + \dfrac{5}{12}\\ \\ &= \dfrac{1}{2} + \dfrac{5}{12}\\ \\ &= \dfrac{6}{12} + \dfrac{5}{12}\\ \\ &= \dfrac{11}{12} \end{align*} :::: :::::::::: ::::::::::{tab-item} d $$ \dfrac{1}{2} \cdot \dfrac{3}{4} - \dfrac{1}{3} \cdot \dfrac{2}{5} $$ ::::{answer} $$ \dfrac{29}{120} $$ :::: ::::{solution} \begin{align*} \dfrac{1}{2} \cdot \dfrac{3}{4} - \dfrac{1}{3} \cdot \dfrac{2}{5} &= \dfrac{3}{8} - \dfrac{2}{15}\\ \\ &= \dfrac{3 \cdot 15}{8 \cdot 15} - \dfrac{2 \cdot 8}{15 \cdot 8}\\ \\ &= \dfrac{45}{120} - \dfrac{16}{120}\\ \\ &= \dfrac{29}{120} \end{align*} :::: :::::::::: ::::::::::: :::::::::::: --- ::::::::::::{exercise} Oppgave 7 --- level: 2 --- Regn ut og forkort svaret så mye som mulig. :::::::::::{tab-set} --- class: tabs-parts --- ::::::::::{tab-item} a $$ 2 \cdot \left(\dfrac{3}{4} + \dfrac{4}{3}\right) : \dfrac{1}{4} $$ ::::{answer} $$ \dfrac{50}{3} $$ :::: ::::{solution} \begin{align*} 2 \cdot \left(\dfrac{3}{4} + \dfrac{4}{3}\right) : \dfrac{1}{4} &= 2 \cdot \left(\dfrac{9}{12} + \dfrac{16}{12}\right) : \dfrac{1}{4}\\ \\ &= 2 \cdot \dfrac{25}{12} : \dfrac{1}{4}\\ \\ &= \dfrac{50}{12} : \dfrac{1}{4}\\ \\ &= \dfrac{50}{12} \cdot \dfrac{4}{1}\\ \\ &= \dfrac{200}{12}\\ \\ &= \dfrac{50}{3} \end{align*} :::: :::::::::: ::::::::::{tab-item} b $$ \dfrac{1}{2} \cdot \left(3 + 4\right) - 2 \cdot \dfrac{1}{4} $$ ::::{answer} $$ 3 $$ :::: ::::{solution} \begin{align*} \dfrac{1}{2} \cdot \left(3 + 4\right) - 2 \cdot \dfrac{1}{4} &= \dfrac{1}{2} \cdot 7 - \dfrac{2}{4}\\ \\ &= \dfrac{7}{2} - \dfrac{1}{2}\\ \\ &= \dfrac{6}{2}\\ \\ &= 3 \end{align*} :::: :::::::::: ::::::::::{tab-item} c $$ \dfrac{2}{3} \cdot \left(\dfrac{3}{4} + \dfrac{1}{2}\right) + \dfrac{1}{5} : \dfrac{1}{3} $$ ::::{answer} $$ \dfrac{43}{30} $$ :::: ::::{solution} \begin{align*} \dfrac{2}{3} \cdot \left(\dfrac{3}{4} + \dfrac{1}{2}\right) + \dfrac{1}{5} : \dfrac{1}{3} &= \dfrac{2}{3} \cdot \left(\dfrac{3}{4} + \dfrac{2}{4}\right) + \dfrac{1}{5} \cdot \dfrac{3}{1}\\ \\ &= \dfrac{2}{3} \cdot \dfrac{5}{4} + \dfrac{3}{5}\\ \\ &= \dfrac{10}{12} + \dfrac{3}{5}\\ \\ &= \dfrac{5}{6} + \dfrac{3}{5}\\ \\ &= \dfrac{25}{30} + \dfrac{18}{30}\\ \\ &= \dfrac{43}{30} \end{align*} :::: :::::::::: ::::::::::{tab-item} d $$ \dfrac{1}{2} \cdot \left(\dfrac{3}{4} - \dfrac{1}{3}\right) + 2 : \dfrac{1}{5} $$ ::::{answer} $$ \dfrac{245}{24} $$ :::: ::::{solution} \begin{align*} \dfrac{1}{2} \cdot \left(\dfrac{3}{4} - \dfrac{1}{3}\right) + 2 : \dfrac{1}{5} &= \dfrac{1}{2} \cdot \left(\dfrac{9}{12} - \dfrac{4}{12}\right) + 2 \cdot \dfrac{5}{1}\\ \\ &= \dfrac{1}{2} \cdot \dfrac{5}{12} + 10\\ \\ &= \dfrac{5}{24} + 10\\ \\ &= \dfrac{5}{24} + \dfrac{240}{24}\\ \\ &= \dfrac{245}{24} \end{align*} :::: :::::::::: ::::::::::: :::::::::::: --- ::::::::::::{exercise} Oppgave 8 --- level: 2 --- Skriv kvadratrøttene så enkelt som mulig. :::::::::::{tab-set} --- class: tabs-parts --- ::::::::::{tab-item} a $$ \sqrt{32} $$ ::::{answer} $$ 4\sqrt{2} $$ :::: ::::{solution} $$ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} $$ :::: :::::::::: ::::::::::{tab-item} b $$ \sqrt{50} $$ ::::{answer} $$ 5\sqrt{2} $$ :::: ::::{solution} $$ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} $$ :::: :::::::::: ::::::::::{tab-item} c $$ \sqrt{27} $$ ::::{answer} $$ 3\sqrt{3} $$ :::: ::::{solution} $$ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} $$ :::: :::::::::: ::::::::::{tab-item} d $$ \sqrt{18} $$ ::::{answer} $$ 3\sqrt{2} $$ :::: ::::{solution} $$ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} $$ :::: :::::::::: ::::::::::: :::::::::::: --- :::::::::::::::{exercise} Oppgave 9 --- level: 2 --- Skriv så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \sqrt{32} + \sqrt{8} $$ ::::{answer} $$ 6\sqrt{2} $$ :::: ::::{solution} \begin{align*} \sqrt{32} + \sqrt{8} &= \sqrt{16 \cdot 2} + \sqrt{4 \cdot 2}\\ \\ &= 4\sqrt{2} + 2\sqrt{2}\\ \\ &= 6\sqrt{2} \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} b $$ \sqrt{50} - \sqrt{2} $$ ::::{answer} $$ 4\sqrt{2} $$ :::: ::::{solution} \begin{align*} \sqrt{50} - \sqrt{2} &= \sqrt{25 \cdot 2} - \sqrt{2}\\ \\ &= 5\sqrt{2} - \sqrt{2}\\ \\ &= 4\sqrt{2} \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} c $$ \sqrt{18} + \sqrt{72} $$ ::::{answer} $$ 9\sqrt{2} $$ :::: ::::{solution} \begin{align*} \sqrt{18} + \sqrt{72} &= \sqrt{9 \cdot 2} + \sqrt{36 \cdot 2}\\ \\ &= 3\sqrt{2} + 6\sqrt{2}\\ \\ &= 9\sqrt{2} \end{align*} :::: ::::::::::::: :::::::::::::{tab-item} d $$ \sqrt{27} - \sqrt{12} $$ ::::{answer} $$ \sqrt{3} $$ :::: ::::{solution} \begin{align*} \sqrt{27} - \sqrt{12} &= \sqrt{9 \cdot 3} - \sqrt{4 \cdot 3}\\ \\ &= 3\sqrt{3} - 2\sqrt{3}\\ \\ &= \sqrt{3} \end{align*} :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 10 --- level: 3 --- Skriv om enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \dfrac{\sqrt{32}}{\sqrt{8}} $$ ::::{answer} $$ 2 $$ :::: ::::{solution} $$ \dfrac{\sqrt{32}}{\sqrt{8}} = \sqrt{\dfrac{32}{8}} = \sqrt{4} = 2 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \dfrac{\sqrt{50}}{\sqrt{2}} $$ ::::{answer} $$ 5 $$ :::: ::::{solution} $$ \dfrac{\sqrt{50}}{\sqrt{2}} = \sqrt{\dfrac{50}{2}} = \sqrt{25} = 5 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{\sqrt{18}}{\sqrt{72}} $$ ::::{answer} $$ \dfrac{1}{2} $$ :::: ::::{solution} $$ \dfrac{\sqrt{18}}{\sqrt{72}} = \sqrt{\dfrac{18}{72}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ \dfrac{\sqrt{27}}{\sqrt{12}} $$ ::::{answer} $$ \dfrac{3}{2} $$ :::: ::::{solution} $$ \dfrac{\sqrt{27}}{\sqrt{12}} = \sqrt{\dfrac{27}{12}} = \sqrt{\dfrac{9}{4}} = \dfrac{3}{2} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Regneregel For to tall $a$ og $b$ gjelder $$ \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} $$ ::: :::::::::::::::{exercise} Oppgave 11 --- level: 3 --- Skriv så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \sqrt{\dfrac{3}{4}} $$ ::::{answer} $$ \dfrac{\sqrt{3}}{2} $$ :::: ::::{solution} $$ \sqrt{\dfrac{3}{4}} = \dfrac{\sqrt{3}}{\sqrt{4}} = \dfrac{\sqrt{3}}{2} $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \sqrt{\dfrac{5}{9}} $$ ::::{answer} $$ \dfrac{\sqrt{5}}{3} $$ :::: ::::{solution} $$ \sqrt{\dfrac{5}{9}} = \dfrac{\sqrt{5}}{\sqrt{9}} = \dfrac{\sqrt{5}}{3} $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \sqrt{\dfrac{8}{25}} $$ ::::{answer} $$ \dfrac{2\sqrt{2}}{5} $$ :::: ::::{solution} $$ \sqrt{\dfrac{8}{25}} = \dfrac{\sqrt{8}}{\sqrt{25}} = \dfrac{\sqrt{4 \cdot 2}}{5} = \dfrac{2\sqrt{2}}{5} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ \sqrt{\dfrac{7}{16}} $$ ::::{answer} $$ \dfrac{\sqrt{7}}{4} $$ :::: ::::{solution} $$ \sqrt{\dfrac{7}{16}} = \dfrac{\sqrt{7}}{\sqrt{16}} = \dfrac{\sqrt{7}}{4} $$ :::: ::::::::::::: :::::::::::::: :::::::::::::::