بسط الجمل العددية الآتية:

$(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})$

$(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})=(\sqrt{5}{)}^2-(\sqrt{3}{)}^2=5-3=2$

تأكد من فهمك

(1)- بسط الجمل العددية الآتية:

$(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})$

$(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})=(\sqrt{5}{)}^2-(\sqrt{3}{)}^2=5-3=2$

$(\sqrt{7}-\sqrt{2}{)}^2$

عملية توزيع قوس على قوس آخر.

$$\begin{gathered} (\sqrt{7}-\sqrt{2}{)}^2=(\sqrt{7}-\sqrt{2}\left)\right(\sqrt{7}-\sqrt{2}) \\ \sqrt{7}\times \sqrt{7}-\sqrt{7}\times \sqrt{2}-\sqrt{2}\times \sqrt{7}+\sqrt{2}\times \sqrt{2}=7-\sqrt{14}-\sqrt{14}+2=9-2\sqrt{14} \end{gathered}$$

طريقة ثانية هي طريقة قانون مربع كامل

$$\begin{gathered} (a\pm b{)}^2=a^2\pm 2ab+b^2 \\ (\sqrt{7}-\sqrt{2}{)}^2=(\sqrt{7}{)}^2-2\times \sqrt{7}\times \sqrt{2}+(\sqrt{2}{)}^2=7-2\sqrt{14}+2=9-2\sqrt{14} \end{gathered}$$

$(\sqrt{125}-\sqrt{20})\left(\sqrt[3]{\frac{8}{27}}\right)$

$(\sqrt{125}-\sqrt{20})\left(\sqrt[3]{\frac{8}{27}}\right)=(5\sqrt{5}-2\sqrt{5})\left(\frac{2}{3}\right)=3\sqrt{5}\left(\frac{2}{3}\right)=\frac{6}{3}\sqrt{5}=2\sqrt{5}$

$\frac{4\sqrt{12}}{5\sqrt[3]{-27}}÷\frac{2\sqrt{24}}{\sqrt{8}}$

$$\begin{gathered} \frac{4\sqrt{12}}{5\sqrt[3]{-27}}÷\frac{2\sqrt{24}}{\sqrt{8}}=\frac{2\times 2\sqrt{3}}{5\times (-3)}÷\frac{2\times 2\sqrt{6}}{2\sqrt{2}}=\frac{4\sqrt{3}}{-15}÷\frac{4\sqrt{3}\times \sqrt{2}}{2\sqrt{2}}=\frac{4\sqrt{3}}{-15}÷\frac{4\sqrt{3}}{2} \\ =\frac{4\sqrt{3}}{-15}\times \frac{2}{4\sqrt{3}}=\frac{-2}{15} \end{gathered}$$

(2)- بسط الجمل العددية التالية واكتب الناتج لأقرب عشر:

$\sqrt{7}(\sqrt{28}-\sqrt{2})-5$

$\begin{array}{rl}& \sqrt{7}(\sqrt{28}-\sqrt{2})-5=\sqrt{7}(2\sqrt{7}-\sqrt{2})-5=\sqrt{7}\times 2\sqrt{7}-\sqrt{7}\times \sqrt{2}-5\\ & =14-\sqrt{14}-5=9-\sqrt{14}=9-3.74=5.26\approx 5.3\end{array}$

$(-125{)}^{\frac{1}{3}}(\frac{1}{10}\sqrt{3}-\frac{1}{4}\sqrt{12})$

$\begin{array}{rl}& (-125{)}^{\frac{1}{3}}(\frac{1}{10}\sqrt{3}-\frac{1}{4}\sqrt{12})=\sqrt[3]{-125}(\frac{1}{10}\sqrt{3}-\frac{1}{4}\times 2\sqrt{3})=-5(\frac{1}{10}\sqrt{3}-\frac{1}{2}\sqrt{3})\\ & =-5\times \frac{1}{10}\sqrt{3}+5\times \frac{1}{2}\sqrt{3}=\frac{-1}{2}\sqrt{3}+\frac{5}{2}\sqrt{3}=\frac{4}{2}\sqrt{3}=2\sqrt{3}=2\times 1.73=3.46=3.5\end{array}$

(3)- بسط الجمل العددية التالية باستعمال تنسيب المقام وترتيب العمليات على الأعداد:

$\frac{1-\sqrt{3}}{4\sqrt{3}}$

$\frac{1-\sqrt{3}}{4\sqrt{3}}=\frac{1-\sqrt{3}}{4\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}(1-\sqrt{3})}{4\sqrt{3}\times \sqrt{3}}=\frac{\sqrt{3}\times 1-\sqrt{3}\times \sqrt{3}}{12}=\frac{\sqrt{3}-3}{12}$

$\frac{1-\sqrt{20}}{\sqrt{5}}$

$\frac{1-\sqrt{20}}{\sqrt{5}}=\frac{1-2\sqrt{5}}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}(1-2\sqrt{5})}{\sqrt{5}\times \sqrt{5}}=\frac{\sqrt{5}\times 1-\sqrt{5}\times 2\sqrt{5}}{5}=\frac{\sqrt{5}-10}{5}$

$\frac{\sqrt{50}-\sqrt{3}}{2\sqrt{3}}-\frac{10-\sqrt{6}}{2\sqrt{6}}$

$\begin{array}{rl}& \frac{\sqrt{50}-\sqrt{3}}{2\sqrt{3}}-\frac{10-\sqrt{6}}{2\sqrt{6}}=\frac{5\sqrt{2}-\sqrt{3}}{2\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}-\frac{10-\sqrt{6}}{2\sqrt{6}}\times \frac{\sqrt{6}}{\sqrt{6}}\\ & =\frac{\sqrt{3}(5\sqrt{2}-\sqrt{3})}{\sqrt{3}\times 2\sqrt{3}}-\frac{\sqrt{6}(10-\sqrt{6})}{\sqrt{6}\times 2\sqrt{6}}=\frac{\sqrt{3}\times 5\sqrt{2}-\sqrt{3}\times \sqrt{3}}{6}-\frac{\sqrt{6}\times 10-\sqrt{6}\times \sqrt{6}}{12}=\frac{5\sqrt{6}-3}{6}-\frac{10\sqrt{6}-6}{12}\\ & =\frac{10\sqrt{6}-6-10\sqrt{6}+6}{12}=\frac{0}{12}=0\end{array}$

(4)- استعمل ترتيب العمليات واكتب الناتج مقرباً إلى مرتبتين عشريتين مستعملاً الحاسبة لكل مما يأتي:

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

$\begin{array}{rl}& {\left(\frac{1}{3}\right)}^2+3^{-3}-3^{\frac{3}{2}}=\frac{1}{3^2}+\frac{1}{3^3}-\sqrt{3^3}=\frac{1}{9}+\frac{1}{27}-\sqrt{27}=\frac{1}{9}+\frac{1}{27}-3\sqrt{3}\\ & =\frac{3+1-81\sqrt{3}}{27}=\frac{4-81\sqrt{3}}{27}=\frac{4-81\times \left(1.73\right)}{27}=\frac{4-140.13}{27}=\frac{-136.13}{27}=-5.042\approx -5.04\end{array}$

$(27{)}^{\frac{1}{2}}-(-9{)}^0+3^2\times (5{)}^{\frac{1}{2}}$

$\begin{array}{rl}& (27{)}^{\frac{1}{2}}-(-9{)}^0+3^2\times (5{)}^{\frac{1}{2}}=\sqrt{27}-1+9\times \sqrt{5}=3\sqrt{3}-1+9\times 2.236\\ & =3\times 1.73-1+20.124=5.19-1+20.124=24.314\approx 24.31\end{array}$

(5)- استعمل الحاسبة لتكتب الناتج بالصورة العلمية للعدد مقرباً لأقرب مرتبتين عشريتين:

$6.43\times {10}^{-5}-0.25\times {10}^{-4}$

$\begin{array}{rl}& 6.43\times {10}^{-5}-0.25\times {10}^{-4}=0.643\times {10}^{-4}-0.25\times {10}^{-4}\\ & =(0.643-0.25)\times {10}^{-4}=0.393\times {10}^{-4}\end{array}$

${(9.23\times {10}^{-3})}^2$

$\begin{array}{rl}{(9.23\times {10}^{-3})}^2& =9.23\times {10}^{-3}\times 9.23\times {10}^{-3}=85.1929\times {10}^{-6}\\ & =8.52\times {10}^{-5}\end{array}$