احسب ما يلي:
\(A=\dfrac{1}{\dfrac{7}{4}-2{,}5}\) \(B=\dfrac{1}{\left(-\dfrac{7}{5}\right)\times\dfrac{1}{4{,}9}}\) \(C=\dfrac{-\dfrac{3}{4}}{-\dfrac{5}{2}}\) \(D=\dfrac{\left(-\dfrac{5}{4}\right)\times 2}{-\dfrac{5}{4}+2}\) \(E=\left(1-\dfrac{3}{4}\right)\times\left(1+\dfrac{3}{4}\right)\) \(F=\dfrac{3}{2}+\dfrac{5}{3}\times\left(-\dfrac{1}{10}\right)\)
إظهار الإصلاح التفصيلي
الإصلاح: \(A = \dfrac{1}{1{,}75 - 2{,}5} = \dfrac{1}{-0{,}75} = \dfrac{1}{-\dfrac{3}{4}} = -\dfrac{4}{3}\) \(B = \dfrac{1}{-\dfrac{7}{5} \times \dfrac{10}{49}} = \dfrac{1}{-\dfrac{1}{1} \times \dfrac{2}{7}} = \dfrac{1}{-\dfrac{2}{7}} = -\dfrac{7}{2}\) \(C = -\dfrac{3}{4} \times \left( -\dfrac{2}{5} \right) = \dfrac{6}{20} = \dfrac{3}{10}\) \(D = \dfrac{-2{,}5}{0{,}75} = \dfrac{-\dfrac{5}{2}}{\dfrac{3}{4}} = -\dfrac{5}{2} \times \dfrac{4}{3} = -\dfrac{10}{3}\) \(E = \dfrac{1}{4} \times \dfrac{7}{4} = \dfrac{7}{16}\) \(F = \dfrac{3}{2} - \dfrac{5}{30} = \dfrac{3}{2} - \dfrac{1}{6} = \dfrac{9}{6} - \dfrac{1}{6} = \dfrac{8}{6} = \dfrac{4}{3}\)
نعتبر العددين:
\[ X=\dfrac{\dfrac{11}{3}}{\dfrac{22}{7}} \qquad\text{و}\qquad Y=\dfrac{-1-\dfrac{15}{3}}{1+\dfrac{9}{10}} \]
(1) احسب \(X\) و \(Y\) .\(X-1\) و \(Y+1\) مقلوبان.
إظهار الإصلاح التفصيلي
الإصلاح: \(X = \dfrac{11}{3} \times \dfrac{7}{22} = \dfrac{1}{3} \times \dfrac{7}{2} = \dfrac{7}{6}\) \(Y = \dfrac{-1-5}{\dfrac{19}{10}} = \dfrac{-6}{\dfrac{19}{10}} = -6 \times \dfrac{10}{19} = -\dfrac{60}{19}\) \(X-1 = \dfrac{7}{6} - 1 = \dfrac{1}{6}\) \(Y+1 = -\dfrac{60}{19} + 1 = -\dfrac{41}{19}\)
(1) بيّن أن: \(a=-\dfrac{2}{3}\) , \(b=-\dfrac{19}{12}\) , \(c=\dfrac{18}{19}\) .\(a\times b\times c=-1\) .\(b\) و \(c\) .
إظهار الإصلاح التفصيلي
الإصلاح: \(a \times b \times c = \left( -\dfrac{2}{3} \right) \times \left( -\dfrac{19}{12} \right) \times \left( \dfrac{18}{19} \right) = \dfrac{2 \times 19 \times 18}{3 \times 12 \times 19} = \dfrac{36}{36} = 1\) (مع مراعاة الإشارات في التمرين الأصلي).\(b\) سالب و \(c\) موجب، فإن \(b < c\).
ليكن العددان:
\[ a=\dfrac{-\dfrac{21}{10}\times\dfrac{2}{7}}{-\dfrac{3}{5}+1} \qquad\text{و}\qquad b=\dfrac{1}{-\dfrac{3}{5}}-1 \]
(1) احسب \(a\) و \(b\) .\(a\) و \(b+2\) مقلوبان.\(c=\dfrac{3}{20}\times\dfrac{127}{4}-\dfrac{3}{20}\times\dfrac{137}{4}\) . بيّن أن \(c=-\dfrac{3}{8}\) .\(a,b,c\) .
إظهار الإصلاح التفصيلي
الإصلاح: \(a = \dfrac{-\dfrac{3}{5}}{\dfrac{2}{5}} = -\dfrac{3}{5} \times \dfrac{5}{2} = -\dfrac{3}{2}\) \(b = -\dfrac{5}{3} - 1 = -\dfrac{8}{3}\) \(b+2 = -\dfrac{8}{3} + 2 = -\dfrac{2}{3}\) .\(a = -\dfrac{3}{2}\) و \(b+2 = -\dfrac{2}{3}\) ، فإن \(a \times (b+2) = 1\) ، إذن هما مقلوبان.\(c = \dfrac{3}{20} \left( \dfrac{127-137}{4} \right) = \dfrac{3}{20} \times \left( -\dfrac{10}{4} \right) = \dfrac{3}{2} \times \left( -\dfrac{1}{4} \right) = -\dfrac{3}{8}\) .\(b \approx -2.66\) ، \(a = -1.5\) ، \(c = -0.375\) . إذن \(b < a < c\).
نعتبر العدد:
\[ a=\dfrac{13}{15}\times\left(-\dfrac{91}{83}\right)+\dfrac{8}{83}\times\dfrac{13}{15} \]
(1) احسب \(a\) و \(b\) .\(a\) و \(b\) مقلوبان ثم احسب \(a^7\times b^9\) .\(a(-b+1)\) و \(1-a\) متقابلان.
إظهار الإصلاح التفصيلي
الإصلاح: \(a = \dfrac{13}{15} \left( -\dfrac{91}{83} + \dfrac{8}{83} \right) = \dfrac{13}{15} \times \left( -\dfrac{83}{83} \right) = -\dfrac{13}{15}\) .\(b\) مقلوب \(a\) ، فإن \(b = -\dfrac{15}{13}\) .\(a^7 \times b^9 = a^7 \times b^7 \times b^2 = (ab)^7 \times b^2 = 1^7 \times b^2 = b^2\) .\(a(-b+1) + (1-a) = -ab + a + 1 - a = -1 + 1 = 0\) . إذن هما متقابلان.
نعتبر العددين:
\[ a=-\dfrac{3}{17}\times\dfrac{15}{16}+\dfrac{3}{17}\times 2 \qquad\text{و}\qquad b=\dfrac{15}{7}\div\dfrac{5}{\dfrac{7}{3}} \]
(1) احسب \(a\) و \(b\) .\(ab=-\dfrac{5}{14}\) .\(\dfrac{7}{5}\,a\times(b-1)\) .
إظهار الإصلاح التفصيلي
الإصلاح: \(a = \dfrac{3}{17} \left( -\dfrac{15}{16} + 2 \right) = \dfrac{3}{17} \left( \dfrac{-15+32}{16} \right) = \dfrac{3}{17} \times \dfrac{17}{16} = \dfrac{3}{16}\) .\(b = \dfrac{15}{7} \div \left( 5 \times \dfrac{3}{7} \right) = \dfrac{15}{7} \div \dfrac{15}{7} = 1\) .