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

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Werk uit en vereenvoudig

1. \((\frac{2}{18}+a)(\frac{-9}{18}+p)\)
2. \((a-\frac{8}{8})(\frac{-12}{12}-q)\)
3. \((a-\frac{-15}{5})(\frac{7}{9}+x)\)
4. \((p-\frac{10}{17})(\frac{-16}{11}+p)\)
5. \((x-\frac{-20}{17})(18-p)\)
6. \((12-p)(\frac{-11}{19}-b)\)
7. \((q-\frac{12}{7})(\frac{20}{19}-a)\)
8. \((p+\frac{-17}{17})(p+\frac{-12}{14})\)
9. \((\frac{14}{2}-b)(b+\frac{-20}{14})\)
10. \((\frac{13}{18}+b)(\frac{-5}{4}+p)\)
11. \((p-\frac{3}{6})(x-\frac{14}{7})\)
12. \((\frac{19}{14}+a)(\frac{-7}{13}+q)\)

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Werk uit en vereenvoudig

Verbetersleutel

1. \(\begin{align}(\frac{2}{18}+a)(\frac{-9}{18}+p)& =(\color{red}{a}+\color{orange}{\frac{1}{9}}).(\color{green}{p}+\color{blue}{\frac{-1}{2}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(p)} + \color{red}{(a)} . \color{blue}{(\frac{-1}{2})} + \color{orange}{(\frac{1}{9})} . \color{green}{(p)} + \color{orange}{(\frac{1}{9})} . \color{blue}{(\frac{-1}{2})}& \text{Boogjes en uitschrijven} \\ & = ap-\frac{1}{2}a+\frac{1}{9}p-\frac{1}{9}.\frac{1}{2}& \text{Uitwerken } \\ & =ap-\frac{1}{2}a+\frac{1}{9}p-\frac{1}{18} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
2. \(\begin{align}(a-\frac{8}{8})(\frac{-12}{12}-q)& =(\color{red}{a}+\color{orange}{(-1)}).(\color{green}{-q}+\color{blue}{(-1)})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(-q)} + \color{red}{(a)} . \color{blue}{(-1)} + \color{orange}{(-1)} . \color{green}{(-q)} + \color{orange}{(-1)} . \color{blue}{(-1)}& \text{Boogjes en uitschrijven} \\ & = -aq-1a+1q+1.1& \text{Uitwerken } \\ & =-aq-1a+1q+1 & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
3. \(\begin{align}(a-\frac{-15}{5})(\frac{7}{9}+x)& =(\color{red}{a}+\color{orange}{3}).(\color{green}{x}+\color{blue}{\frac{7}{9}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(x)} + \color{red}{(a)} . \color{blue}{(\frac{7}{9})} + \color{orange}{(3)} . \color{green}{(x)} + \color{orange}{(3)} . \color{blue}{(\frac{7}{9})}& \text{Boogjes en uitschrijven} \\ & = ax+\frac{7}{9}a+3x+3.\frac{7}{9}& \text{Uitwerken } \\ & =ax+\frac{7}{9}a+3x+\frac{21}{9} & \text{Uitrekenen (Geen Smoothie!)} \\ & = ax+\frac{7}{9}a+3x+\frac{7}{3} & \text{Vereenvoudig} \\\end{align}\)
4. \(\begin{align}(p-\frac{10}{17})(\frac{-16}{11}+p)& =(\color{red}{p}+\color{orange}{\frac{-10}{17}}).(\color{green}{p}+\color{blue}{\frac{-16}{11}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(p)} + \color{red}{(p)} . \color{blue}{(\frac{-16}{11})} + \color{orange}{(\frac{-10}{17})} . \color{green}{(p)} + \color{orange}{(\frac{-10}{17})} . \color{blue}{(\frac{-16}{11})}& \text{Boogjes en uitschrijven} \\ & = pp-\frac{16}{11}p-\frac{10}{17}p+\frac{10}{17}.\frac{16}{11}& \text{Uitwerken } \\ & =p^2+((\frac{-16}{11})+(\frac{-10}{17}))p+\frac{160}{187} & \text{Neem samen} \\ &=p^2-\frac{382}{187}p+\frac{160}{187} & \text{Maak de som} \\\end{align}\)
5. \(\begin{align}(x-\frac{-20}{17})(18-p)& =(\color{red}{x}+\color{orange}{\frac{20}{17}}).(\color{green}{-p}+\color{blue}{18})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(-p)} + \color{red}{(x)} . \color{blue}{(18)} + \color{orange}{(\frac{20}{17})} . \color{green}{(-p)} + \color{orange}{(\frac{20}{17})} . \color{blue}{(18)}& \text{Boogjes en uitschrijven} \\ & = -px+18x-\frac{20}{17}p+\frac{20}{17}.18& \text{Uitwerken } \\ & =-px+18x-\frac{20}{17}p+\frac{360}{17} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
6. \(\begin{align}(12-p)(\frac{-11}{19}-b)& =(\color{red}{-p}+\color{orange}{12}).(\color{green}{-b}+\color{blue}{\frac{-11}{19}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-p)} . \color{green}{(-b)} + \color{red}{(-p)} . \color{blue}{(\frac{-11}{19})} + \color{orange}{(12)} . \color{green}{(-b)} + \color{orange}{(12)} . \color{blue}{(\frac{-11}{19})}& \text{Boogjes en uitschrijven} \\ & = bp+\frac{11}{19}p-12b-12.\frac{11}{19}& \text{Uitwerken } \\ & =bp+\frac{11}{19}p-12b-\frac{132}{19} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
7. \(\begin{align}(q-\frac{12}{7})(\frac{20}{19}-a)& =(\color{red}{q}+\color{orange}{\frac{-12}{7}}).(\color{green}{-a}+\color{blue}{\frac{20}{19}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(-a)} + \color{red}{(q)} . \color{blue}{(\frac{20}{19})} + \color{orange}{(\frac{-12}{7})} . \color{green}{(-a)} + \color{orange}{(\frac{-12}{7})} . \color{blue}{(\frac{20}{19})}& \text{Boogjes en uitschrijven} \\ & = -aq+\frac{20}{19}q+\frac{12}{7}a-\frac{12}{7}.\frac{20}{19}& \text{Uitwerken } \\ & =-aq+\frac{20}{19}q+\frac{12}{7}a-\frac{240}{133} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
8. \(\begin{align}(p+\frac{-17}{17})(p+\frac{-12}{14})& =(\color{red}{p}+\color{orange}{(-1)}).(\color{green}{p}+\color{blue}{\frac{-6}{7}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(p)} + \color{red}{(p)} . \color{blue}{(\frac{-6}{7})} + \color{orange}{(-1)} . \color{green}{(p)} + \color{orange}{(-1)} . \color{blue}{(\frac{-6}{7})}& \text{Boogjes en uitschrijven} \\ & = pp-\frac{6}{7}p-1p+1.\frac{6}{7}& \text{Uitwerken } \\ & =p^2+((\frac{-6}{7})+(-1))p+\frac{6}{7} & \text{Neem samen} \\ &=p^2-\frac{13}{7}p+\frac{6}{7} & \text{Maak de som} \\\end{align}\)
9. \(\begin{align}(\frac{14}{2}-b)(b+\frac{-20}{14})& =(\color{red}{-b}+\color{orange}{7}).(\color{green}{b}+\color{blue}{\frac{-10}{7}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-b)} . \color{green}{(b)} + \color{red}{(-b)} . \color{blue}{(\frac{-10}{7})} + \color{orange}{(7)} . \color{green}{(b)} + \color{orange}{(7)} . \color{blue}{(\frac{-10}{7})}& \text{Boogjes en uitschrijven} \\ & = -bb+\frac{10}{7}b+7b-7.\frac{10}{7}& \text{Uitwerken } \\ & =-b^2+(\frac{10}{7}+7)b-\frac{70}{7} & \text{Neem samen} \\ &=-b^2+\frac{59}{7}b-\frac{70}{7} & \text{Maak de som} \\ &=-b^2+\frac{59}{7}b-10 & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)
10. \(\begin{align}(\frac{13}{18}+b)(\frac{-5}{4}+p)& =(\color{red}{b}+\color{orange}{\frac{13}{18}}).(\color{green}{p}+\color{blue}{\frac{-5}{4}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(p)} + \color{red}{(b)} . \color{blue}{(\frac{-5}{4})} + \color{orange}{(\frac{13}{18})} . \color{green}{(p)} + \color{orange}{(\frac{13}{18})} . \color{blue}{(\frac{-5}{4})}& \text{Boogjes en uitschrijven} \\ & = bp-\frac{5}{4}b+\frac{13}{18}p-\frac{13}{18}.\frac{5}{4}& \text{Uitwerken } \\ & =bp-\frac{5}{4}b+\frac{13}{18}p-\frac{65}{72} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
11. \(\begin{align}(p-\frac{3}{6})(x-\frac{14}{7})& =(\color{red}{p}+\color{orange}{\frac{-1}{2}}).(\color{green}{x}+\color{blue}{(-2)})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(x)} + \color{red}{(p)} . \color{blue}{(-2)} + \color{orange}{(\frac{-1}{2})} . \color{green}{(x)} + \color{orange}{(\frac{-1}{2})} . \color{blue}{(-2)}& \text{Boogjes en uitschrijven} \\ & = px-2p-\frac{1}{2}x+\frac{1}{2}.2& \text{Uitwerken } \\ & =px-2p-\frac{1}{2}x+\frac{2}{2} & \text{Uitrekenen (Geen Smoothie!)} \\ & = px-2p-\frac{1}{2}x+1 & \text{Vereenvoudig} \\\end{align}\)
12. \(\begin{align}(\frac{19}{14}+a)(\frac{-7}{13}+q)& =(\color{red}{a}+\color{orange}{\frac{19}{14}}).(\color{green}{q}+\color{blue}{\frac{-7}{13}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(q)} + \color{red}{(a)} . \color{blue}{(\frac{-7}{13})} + \color{orange}{(\frac{19}{14})} . \color{green}{(q)} + \color{orange}{(\frac{19}{14})} . \color{blue}{(\frac{-7}{13})}& \text{Boogjes en uitschrijven} \\ & = aq-\frac{7}{13}a+\frac{19}{14}q-\frac{19}{14}.\frac{7}{13}& \text{Uitwerken } \\ & =aq-\frac{7}{13}a+\frac{19}{14}q-\frac{133}{182} & \text{Uitrekenen (Geen Smoothie!)} \\ & = aq-\frac{7}{13}a+\frac{19}{14}q-\frac{19}{26} & \text{Vereenvoudig} \\\end{align}\)