Distributiviteit \((a+r)(b+s)\)

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Werk uit en vereenvoudig

1. \((q+(-18))(q+\frac{8}{6})\)
2. \((b+\frac{-16}{12})(17-a)\)
3. \(((-16)+q)(\frac{-1}{7}-p)\)
4. \((\frac{2}{5}-q)(\frac{-18}{7}+p)\)
5. \((\frac{-8}{2}+q)(b+\frac{-3}{3})\)
6. \((a+\frac{8}{11})(\frac{9}{9}+p)\)
7. \((a-\frac{-8}{10})(b+\frac{10}{11})\)
8. \((\frac{-7}{5}+p)(\frac{-2}{2}+q)\)
9. \((\frac{20}{6}+b)(7-q)\)
10. \((\frac{13}{12}+q)(a-\frac{11}{10})\)
11. \((\frac{4}{20}+a)(\frac{-18}{11}-p)\)
12. \((\frac{10}{18}-q)(q+\frac{-1}{10})\)

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Werk uit en vereenvoudig

Verbetersleutel

1. \(\begin{align}(q+(-18))(q+\frac{8}{6})& =(\color{red}{q}+\color{orange}{(-18)}).(\color{green}{q}+\color{blue}{\frac{4}{3}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(q)} + \color{red}{(q)} . \color{blue}{(\frac{4}{3})} + \color{orange}{(-18)} . \color{green}{(q)} + \color{orange}{(-18)} . \color{blue}{(\frac{4}{3})}& \text{Boogjes en uitschrijven} \\ & = qq+\frac{4}{3}q-18q-18.\frac{4}{3}& \text{Uitwerken } \\ & =q^2+(\frac{4}{3}+(-18))q-(\frac{-72}{3}) & \text{Neem samen} \\ &=q^2-\frac{50}{3}q-(\frac{-72}{3}) & \text{Maak de som} \\ &=q^2-\frac{50}{3}q+24 & \text{Vereenvoudig indien mogelijk 1} \\\end{align}\)
2. \(\begin{align}(b+\frac{-16}{12})(17-a)& =(\color{red}{b}+\color{orange}{\frac{-4}{3}}).(\color{green}{-a}+\color{blue}{17})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(-a)} + \color{red}{(b)} . \color{blue}{(17)} + \color{orange}{(\frac{-4}{3})} . \color{green}{(-a)} + \color{orange}{(\frac{-4}{3})} . \color{blue}{(17)}& \text{Boogjes en uitschrijven} \\ & = -ab+17b+\frac{4}{3}a-\frac{4}{3}.17& \text{Uitwerken } \\ & =-ab+17b+\frac{4}{3}a-\frac{68}{3} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
3. \(\begin{align}((-16)+q)(\frac{-1}{7}-p)& =(\color{red}{q}+\color{orange}{(-16)}).(\color{green}{-p}+\color{blue}{\frac{-1}{7}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(-p)} + \color{red}{(q)} . \color{blue}{(\frac{-1}{7})} + \color{orange}{(-16)} . \color{green}{(-p)} + \color{orange}{(-16)} . \color{blue}{(\frac{-1}{7})}& \text{Boogjes en uitschrijven} \\ & = -pq-\frac{1}{7}q+16p+16.\frac{1}{7}& \text{Uitwerken } \\ & =-pq-\frac{1}{7}q+16p+\frac{16}{7} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
4. \(\begin{align}(\frac{2}{5}-q)(\frac{-18}{7}+p)& =(\color{red}{-q}+\color{orange}{\frac{2}{5}}).(\color{green}{p}+\color{blue}{\frac{-18}{7}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-q)} . \color{green}{(p)} + \color{red}{(-q)} . \color{blue}{(\frac{-18}{7})} + \color{orange}{(\frac{2}{5})} . \color{green}{(p)} + \color{orange}{(\frac{2}{5})} . \color{blue}{(\frac{-18}{7})}& \text{Boogjes en uitschrijven} \\ & = -pq+\frac{18}{7}q+\frac{2}{5}p-\frac{2}{5}.\frac{18}{7}& \text{Uitwerken } \\ & =-pq+\frac{18}{7}q+\frac{2}{5}p-\frac{36}{35} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
5. \(\begin{align}(\frac{-8}{2}+q)(b+\frac{-3}{3})& =(\color{red}{q}+\color{orange}{(-4)}).(\color{green}{b}+\color{blue}{(-1)})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(b)} + \color{red}{(q)} . \color{blue}{(-1)} + \color{orange}{(-4)} . \color{green}{(b)} + \color{orange}{(-4)} . \color{blue}{(-1)}& \text{Boogjes en uitschrijven} \\ & = bq-1q-4b+4.1& \text{Uitwerken } \\ & =bq-1q-4b+4 & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
6. \(\begin{align}(a+\frac{8}{11})(\frac{9}{9}+p)& =(\color{red}{a}+\color{orange}{\frac{8}{11}}).(\color{green}{p}+\color{blue}{1})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(p)} + \color{red}{(a)} . \color{blue}{(1)} + \color{orange}{(\frac{8}{11})} . \color{green}{(p)} + \color{orange}{(\frac{8}{11})} . \color{blue}{(1)}& \text{Boogjes en uitschrijven} \\ & = ap+1a+\frac{8}{11}p+\frac{8}{11}.1& \text{Uitwerken } \\ & =ap+1a+\frac{8}{11}p+\frac{8}{11} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
7. \(\begin{align}(a-\frac{-8}{10})(b+\frac{10}{11})& =(\color{red}{a}+\color{orange}{\frac{4}{5}}).(\color{green}{b}+\color{blue}{\frac{10}{11}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(b)} + \color{red}{(a)} . \color{blue}{(\frac{10}{11})} + \color{orange}{(\frac{4}{5})} . \color{green}{(b)} + \color{orange}{(\frac{4}{5})} . \color{blue}{(\frac{10}{11})}& \text{Boogjes en uitschrijven} \\ & = ab+\frac{10}{11}a+\frac{4}{5}b+\frac{4}{5}.\frac{10}{11}& \text{Uitwerken } \\ & =ab+\frac{10}{11}a+\frac{4}{5}b+\frac{40}{55} & \text{Uitrekenen (Geen Smoothie!)} \\ & = ab+\frac{10}{11}a+\frac{4}{5}b+\frac{8}{11} & \text{Vereenvoudig} \\\end{align}\)
8. \(\begin{align}(\frac{-7}{5}+p)(\frac{-2}{2}+q)& =(\color{red}{p}+\color{orange}{\frac{-7}{5}}).(\color{green}{q}+\color{blue}{(-1)})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(q)} + \color{red}{(p)} . \color{blue}{(-1)} + \color{orange}{(\frac{-7}{5})} . \color{green}{(q)} + \color{orange}{(\frac{-7}{5})} . \color{blue}{(-1)}& \text{Boogjes en uitschrijven} \\ & = pq-1p-\frac{7}{5}q+\frac{7}{5}.1& \text{Uitwerken } \\ & =pq-1p-\frac{7}{5}q+\frac{7}{5} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
9. \(\begin{align}(\frac{20}{6}+b)(7-q)& =(\color{red}{b}+\color{orange}{\frac{10}{3}}).(\color{green}{-q}+\color{blue}{7})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(-q)} + \color{red}{(b)} . \color{blue}{(7)} + \color{orange}{(\frac{10}{3})} . \color{green}{(-q)} + \color{orange}{(\frac{10}{3})} . \color{blue}{(7)}& \text{Boogjes en uitschrijven} \\ & = -bq+7b-\frac{10}{3}q+\frac{10}{3}.7& \text{Uitwerken } \\ & =-bq+7b-\frac{10}{3}q+\frac{70}{3} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
10. \(\begin{align}(\frac{13}{12}+q)(a-\frac{11}{10})& =(\color{red}{q}+\color{orange}{\frac{13}{12}}).(\color{green}{a}+\color{blue}{\frac{-11}{10}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(a)} + \color{red}{(q)} . \color{blue}{(\frac{-11}{10})} + \color{orange}{(\frac{13}{12})} . \color{green}{(a)} + \color{orange}{(\frac{13}{12})} . \color{blue}{(\frac{-11}{10})}& \text{Boogjes en uitschrijven} \\ & = aq-\frac{11}{10}q+\frac{13}{12}a-\frac{13}{12}.\frac{11}{10}& \text{Uitwerken } \\ & =aq-\frac{11}{10}q+\frac{13}{12}a-\frac{143}{120} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
11. \(\begin{align}(\frac{4}{20}+a)(\frac{-18}{11}-p)& =(\color{red}{a}+\color{orange}{\frac{1}{5}}).(\color{green}{-p}+\color{blue}{\frac{-18}{11}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(-p)} + \color{red}{(a)} . \color{blue}{(\frac{-18}{11})} + \color{orange}{(\frac{1}{5})} . \color{green}{(-p)} + \color{orange}{(\frac{1}{5})} . \color{blue}{(\frac{-18}{11})}& \text{Boogjes en uitschrijven} \\ & = -ap-\frac{18}{11}a-\frac{1}{5}p-\frac{1}{5}.\frac{18}{11}& \text{Uitwerken } \\ & =-ap-\frac{18}{11}a-\frac{1}{5}p-\frac{18}{55} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
12. \(\begin{align}(\frac{10}{18}-q)(q+\frac{-1}{10})& =(\color{red}{-q}+\color{orange}{\frac{5}{9}}).(\color{green}{q}+\color{blue}{\frac{-1}{10}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-q)} . \color{green}{(q)} + \color{red}{(-q)} . \color{blue}{(\frac{-1}{10})} + \color{orange}{(\frac{5}{9})} . \color{green}{(q)} + \color{orange}{(\frac{5}{9})} . \color{blue}{(\frac{-1}{10})}& \text{Boogjes en uitschrijven} \\ & = -qq+\frac{1}{10}q+\frac{5}{9}q-\frac{5}{9}.\frac{1}{10}& \text{Uitwerken } \\ & =-q^2+(\frac{1}{10}+\frac{5}{9})q-\frac{5}{90} & \text{Neem samen} \\ &=-q^2+\frac{59}{90}q-\frac{5}{90} & \text{Maak de som} \\ &=-q^2+\frac{59}{90}q-\frac{1}{18} & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)