استعمل ترتيب العمليات على الأعداد النسبية لتكتب كل مقدار بأبسط صورة:

$6.03\times 1.5-4.03\times 0.9=$

$\begin{array}{rl}& (6.03\times 1.5)-(4.03\times 0.9)\\ & =9.045-3.627=5.418\end{array}$

اختبار الفصل

استعمل ترتيب العمليات على الأعداد النسبية لتكتب كل مقدار بأبسط صورة:

$\frac{-8}{22}\times \frac{11}{-4}+\frac{1}{2}\times \frac{11}{4}=\dots \dots$

$\begin{array}{rl}& (\frac{-8}{22}\times \frac{11}{-4})+(\frac{1}{2}\times \frac{11}{4})\\ & =1+\frac{11}{8}=\frac{19}{8}\end{array}$

$\frac{9}{5}÷\frac{-22}{25}-\frac{9}{4}\times$

$\begin{array}{rl}& (\frac{9}{2}÷\frac{25}{-22})-(\frac{9}{4}÷\frac{3}{-16})\\ & =-\frac{225}{44}+\frac{27}{64}=\frac{-3600+297}{704}\\ & =\frac{-3303}{704}\end{array}$

$\frac{7}{4}-\frac{-5}{12}\times \frac{48}{15}÷\frac{8}{21}=\dots \dots$

$\begin{array}{rl}& \frac{7}{4}+(\frac{5}{12}\times \frac{48}{15})\times \frac{21}{8}\\ & =\frac{7}{4}+\frac{7}{2}=\frac{7+14}{4}=\frac{21}{4}\end{array}$

$6.03\times 1.5-4.03\times 0.9=$

$\begin{array}{rl}& (6.03\times 1.5)-(4.03\times 0.9)\\ & =9.045-3.627=5.418\end{array}$

استعمل ترتيب العمليات لتحسب كلاً مما يأتي:

$\frac{1}{3}+3^{-2}+(-1{)}^{-4}\times \frac{1}{6}=\dots \dots$

$\begin{array}{rl}& \frac{1}{3}+\frac{1}{3^2}+\frac{1}{(-1{)}^4}\times \frac{1}{6}\\ & =\frac{1}{3}+\frac{1}{9}+\frac{1}{6}\\ & =\frac{6+2+3}{18}=\frac{11}{18}\end{array}$

$(-3{)}^{-3}-(8{)}^0+1^3+\frac{1}{9}=\dots \dots$

$\begin{array}{rl}& \frac{1}{(-3{)}^3}+1+1+\frac{1}{9}\\ & =-\frac{1}{27}+\frac{1}{9}=\frac{-1+3}{27}\\ & =\frac{2}{27}\end{array}$

$4^{-2}-2^4-(-5{)}^2-\frac{1}{5}=\dots \dots$

$\begin{array}{rl}& \frac{1}{4^2}-16-25-\frac{1}{5}\\ & =\frac{1}{16}-\frac{41}{1}-\frac{1}{5}\\ & =\frac{5-3280-16}{480}=\frac{-3291}{80}\end{array}$

$(-4{)}^0-(-6{)}^{-2}+1^{-4}-\frac{1}{6}=\dots \dots$

$\begin{array}{rl}& =1-\frac{1}{(-6{)}^2}+1-\frac{1}{6}\\ & 1-\frac{1}{36}+1-\frac{1}{6}\\ & =2-\frac{1}{36}-\frac{1}{6}\\ & =\frac{72-1-6}{36}=\frac{65}{36}\end{array}$

اكتب الأعداد التالية بالصورة الرقمية:

$3.4\times {10}^4=\dots \dots$

$3.4\times {10}^4=34000$

$2.14\times {10}^{-5}=\dots \dots$

$2.14\times {10}^{-5}=0.0000214$

$8.05\times {10}^{-6}=\dots \dots$

$8.05\times {10}^{-6}=0.00000805$

اكتب الأعداد التالية بالصورة العلمية:

$720000=\dots \dots$

$720000=7.2\times {10}^5$

$0.00012=\dots \dots$

$0.00012=1.2\times {10}^{-4}$

$0.0000066=\dots \dots$

$0.0000066=6.6\times {10}^{-6}$

بسط واكتب ناتج الضرب بوصفه قوة واحدة:

$5^{-6}\times 5^2=\dots \dots$

$5^{-6}\times 5^2=5^{-6+2}\times 5^{-4}$

$(-1{)}^{-4}\times 1^{-3}=\dots \dots$

$\frac{t}{(-1{)}^4}\times \frac{t}{t^3}=1\times 1=1$

$9^{-5}\times {\left(\frac{1}{3}\right)}^5=\dots \dots$

$\begin{array}{rl}& {\left(3^2\right)}^{-5}\times {\left(3^{-1}\right)}^5\\ & =3^{-10}\times 3^{-5}\\ & =3^{-10-5}=3^{-15}\end{array}$

${\left(\frac{1}{2}\right)}^{-1}\times {\left(\frac{1}{2}\right)}^{-3}\times {\left(\frac{1}{2}\right)}^6=\dots \dots$

${\left(\frac{1}{2}\right)}^{-1-3+6}={\left(\frac{1}{2}\right)}^2=2^{-2}$

${12}^{-3}\times (-3{)}^4\times (3{)}^5=\dots \dots$

$\begin{array}{rl}& (2^2\times 3^{-3}\times 3^4\times 3^5\\ & =2^{-6}\times 3^{-3}\times 3^4\times 3^5\\ & ={12}^{-6}\times 3^{-3+4+5}=2^{-6}\times 3^6\end{array}$

بسط واكتب الناتج بوصفه قوة واحدة:

$\frac{5^{-4}}{5^9}=\dots \dots$

$\frac{5^{-4}}{5^{-9}}=5^{-4}\times 5^{-9}=5^{-13}$

${\left(7^{-3}\right)}^{-2}=\dots \dots$

${\left(7^{-3}\right)}^{-2}=7^{-3\times (-2)}=7^6$

$\frac{50\times 5^{-6}}{5^3\times 8}=\dots \dots$

$\begin{array}{rl}& \frac{50\times 5^{-6}}{5^3\times 8}=\frac{5^2\times 2\times 5^{-6}}{5^3\times 2^3}\\ & =\frac{5^{2+(-6)-3}}{2^{3-1}}=\frac{1}{2^2\times 5^7}\end{array}$

$\frac{(-3{)}^4\times 3^3\times (-9{)}^5}{3^{-6}\times {(-3^4)}^3}=\dots \dots$

$\begin{array}{rl}& \frac{3^4\times 3^3\times {(-3^2)}^5}{5^{-6}\times (-3^{12})}\\ & =\frac{3^7\times 3^{10}}{-3^{-6}\times 3^{12}}=\frac{3^{17}}{-3^6}=3^{17-6}=3^{11}\end{array}$

$\frac{(-8{)}^3\times 2^{-2}\times (-4^{-2})}{2^{-7}\times {\left(2^3\right)}^3}=\dots \dots$

$\begin{array}{rl}& \frac{(-8{)}^3\times 2^{-2}\times (-4^{-2})}{2^{-7}\times {\left(2^3\right)}^3}\\ & =\frac{{(-2^3)}^3\times 2^{-2}\left({(-2^2)}^{-2}\right)}{2^{-7}\times 2^9}\\ & =\frac{+2^9\times 2^2\times 2^{-4}}{2^{-7}\times 2^9}\\ & =2^{9+(-2)+(-4)-(-7)-9}\\ & =2^{9-2-4+7-9}=2^3\end{array}$

استعمل الحاسبة لتكتب الناتج على صورة كسر عشري:

$\frac{1}{5}+\frac{3}{10}=\dots \dots$

$\frac{1}{5}+\frac{3}{10}=\frac{2+3}{10}=\frac{5}{10}=\frac{1}{2}=0.5$

$\frac{2}{3}-\frac{5}{6}=\dots \dots$

$\begin{array}{rl}& \frac{2}{3}-\frac{5}{6}=\frac{4-5}{6}=\frac{-1}{6}\\ & =0.1666-=0.1\overline{6}\end{array}$

$\frac{2}{7}-\frac{1}{2}+\frac{2}{3}=\dots \dots$

$\begin{array}{rl}& \frac{2}{7}-\frac{1}{2}+\frac{2}{3}=\frac{12-21+28}{42}\\ & =\frac{19}{42}=0.452\end{array}$

$\frac{1}{6}\times \frac{48}{3}=\dots \dots$

$\frac{1}{6}\times \frac{48}{3}=\frac{8}{3}=2.666\dots ..=2.\overline{6}$

$\frac{2}{-7}÷\frac{18}{35}=\dots \dots$

$\begin{array}{rl}& \frac{2}{-7}÷\frac{18}{35}=\frac{2}{-7}\times \frac{35}{18}\\ & =-\frac{5}{9}=-0.555\dots =-0.5\end{array}$

$\frac{3}{5}\times \frac{7}{15}÷\frac{14}{35}=\dots \dots$

$\begin{array}{rl}& \frac{3}{5}\times \frac{7}{15}÷\frac{14}{35}=(\frac{3}{5}\times \frac{7}{15})\times \frac{35}{14}\\ & =\frac{3\times 7}{5\times 15}\times \frac{35}{14}=\frac{7}{10}=0.7\end{array}$

استعمل الحاسبة لتكتب ناتج الضرب والقسمة بالصورة العلمية للعدد:

$(3.5\times {10}^{-6})(8.1\times {10}^{-3})=\dots \dots$

$\begin{array}{rl}& (3.5\times 8.1)({10}^{-6}\times {10}^{-3})\\ & =28.35\times {10}^{-9}\\ & =2.835\times {10}^{-8}\end{array}$

$\frac{7.29\times {10}^7}{0.09\times {10}^3}=\dots \dots$

$\begin{array}{rl}& (7.29÷0.09)({10}^7\times {10}^3)\\ & =81\times {10}^4\\ & =8.1\times {10}^5\end{array}$

بسط الجمل العددية الكسرية التالية واكتب الناتج بأبسط صورة:

$\frac{\sqrt[3]{-27}}{5}\times \frac{|-35|}{9}+\frac{2}{28}\times \frac{-7}{\sqrt[3]{8}}=\dots \dots$

$\begin{array}{rl}& (\frac{-3}{5}\times \frac{35}{9})+(\frac{2}{28}\times \frac{-7}{2})\\ & =\frac{-7}{3}+\frac{-1}{4}=\frac{-28-3}{12}=\frac{-31}{12}\end{array}$

$\frac{6}{\sqrt{49}}÷\frac{|-54|}{-7}-\frac{\sqrt[3]{-125}}{3}\times \frac{-15}{6}=\dots \dots$

$\begin{array}{rl}& (\frac{6}{7}÷\frac{54}{7})-(\frac{-5}{3}\times \frac{-15}{6})\\ & =(\frac{6}{7}\times \frac{7}{54})-\frac{25}{6}\\ & =\frac{1}{9}-\frac{25}{6}\\ & =\frac{2-75}{18}=\frac{-37}{18}\end{array}$