اختبار تفاعلي شامل 📝 في الجبر والحساب

التصحيح المفصّل للعبارة $g$:

$$g = \left[\left(2\sqrt{2} - \sqrt{17}\right) \times \left(2\sqrt{2} + \sqrt{17}\right)\right]^{153} \times \left(2\sqrt{2} + \sqrt{17}\right)$$

$$g = \left[\left(2\sqrt{2}\right)^2 - \left(\sqrt{17}\right)^2\right]^{153} \times \left(2\sqrt{2} + \sqrt{17}\right)$$

$$g = \left[8 - 17\right]^{153} \times \left(2\sqrt{2} + \sqrt{17}\right)$$

$$g = \left[-9\right]^{153} \times \left(2\sqrt{2} + \sqrt{17}\right)$$

التصحيح المفصّل للعبارة $h$:

$$h = \frac{1}{2^4} \times 2^3 \times (\sqrt{2})^3$$

$$h = \frac{1}{16} \times 8 \times 2\sqrt{2}$$

$$h = \frac{16\sqrt{2}}{16}$$

$$h = \sqrt{2}$$

**ملاحظة:** إذا كانت $(\sqrt{2})^3 = 2\sqrt{2}$ فإن: $$h = \frac{1}{16} \times 8 \times 2\sqrt{2} = \frac{16\sqrt{2}}{16} = \sqrt{2}$$ (النتيجة الصحيحة هي **sqrt(2)** وليس sqrt(2)/2 كما حسبت سابقاً. سأعتمد **sqrt(2)** كإجابة متوقعة. **التعديل في الكود تم**)

التصحيح المفصّل للعبارة $k$:

$$k = \sqrt{3 \times 27} - 1$$

$$k = \sqrt{81} - 1$$

$$k = 9 - 1$$

$$k = 8$$

التصحيح المفصّل للعبارة $l$:

$$l = \frac{27}{2} - 4 \times \frac{1}{8} \times \frac{1}{27}$$

$$l = \frac{27}{2} - \frac{1}{2} \times \frac{1}{27}$$

$$l = \frac{27}{2} - \frac{1}{54}$$

$$l = \frac{27 \times 27}{2 \times 27} - \frac{1}{54}$$

$$l = \frac{729 - 1}{54}$$

$$l = \frac{728}{54}$$

$$l = \frac{364}{27}$$

التصحيح المفصّل للعبارة $m$:

$$m = \frac{5^1 \times 5^2 \times (3 \times 5)^2}{3^3\times 5^3}$$

$$m = \frac{5^{1+2} \times 3^2 \times 5^2}{3^3\times 5^3}$$

$$m = \frac{3^2 \times 5^5}{3^3\times 5^3}$$

$$m = 3^{2-3} \times 5^{5-3}$$

$$m = 3^{-1} \times 5^2$$

$$m = \frac{25}{3}$$

التصحيح المفصّل للعبارة $n$:

$$n = \sqrt{9 \times 3} - \sqrt{36 \times 2}$$

$$n = 3\sqrt{3} - 6\sqrt{2}$$

التصحيح المفصّل للعبارة $p$:

$$p = \sqrt{\frac{3}{2} \times \frac{2}{3}}$$

$$p = \sqrt{1}$$

$$p = 1$$

التصحيح المفصّل للعبارة $q$:

$$q = \left(\frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{2}}\right)^3 \times 2^{-3}$$

$$q = \left(1\right)^3 \times \frac{1}{2^3}$$

$$q = 1 \times \frac{1}{8}$$

$$q = \frac{1}{8}$$

التصحيح المفصّل للعبارة $r$:

$$r = \frac{1}{2}\left[\frac{1}{2}\left(\frac{5}{2}\right)^{2}\right]^{-1} \times (9 \times 5)$$

$$r = \frac{1}{2}\left[\frac{1}{2} \times \frac{25}{4}\right]^{-1} \times 45$$

$$r = \frac{1}{2}\left[\frac{25}{8}\right]^{-1} \times 45$$

$$r = \frac{1}{2} \times \frac{8}{25} \times 45$$

$$r = \frac{4}{25} \times 45$$

$$r = \frac{4 \times 9}{5}$$

$$r = \frac{36}{5}$$

التصحيح المفصّل للعبارة $D$:

$$D = (7 \times 4) \times (10^{12} \times 10^4)$$

$$D = 28 \times 10^{12+4}$$

$$D = 28 \times 10^{16}$$

$$D = 2.8 \times 10^1 \times 10^{16}$$

$$D = 2.8 \times 10^{17}$$

التصحيح المفصّل للعبارة $x$:

$$x = \sqrt{5} - (\sqrt{5} \times \sqrt{5} + \sqrt{5} \times 1)$$

$$x = \sqrt{5} - (5 + \sqrt{5})$$

$$x = \sqrt{5} - 5 - \sqrt{5}$$

$$x = -5$$

التصحيح المفصّل للعبارة $y$:

$$y = \left(\frac{2}{5}\right)^2 + 2 \times \left(\frac{3}{5}\right)^3 + \frac{1}{3} \times \frac{1}{25}$$

$$y = \frac{4}{25} + 2 \times \frac{27}{125} + \frac{1}{75}$$

$$y = \frac{4}{25} + \frac{54}{125} + \frac{1}{75}$$

$$y = \frac{4 \times 15}{375} + \frac{54 \times 3}{375} + \frac{1 \times 5}{375}$$

$$y = \frac{60 + 162 + 5}{375}$$

$$y = \frac{227}{375}$$

التصحيح المفصّل للعبارة $z$:

$$z = \frac{2\sqrt{5}}{\sqrt{3}} - \frac{\sqrt{3}}{\sqrt{3}}$$

$$z = \frac{2\sqrt{5} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} - 1$$

$$z = \frac{2\sqrt{15}}{3} - 1$$

حساب $A$:

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

$$A = \left[\left(\frac{1}{2}\right)^5\right]^{-1}$$

$$A = \left(\frac{1}{2}\right)^{-5}$$

$$A = 2^5$$

$$A = 32$$

حساب $B$:

$$B = \frac{1}{27} \times \frac{1}{9} \times \left[\frac{2^3}{3^3} \times \frac{5}{9}\right]^{-1}$$

$$B = \frac{1}{243} \times \left[\frac{8}{27} \times \frac{5}{9}\right]^{-1}$$

$$B = \frac{1}{243} \times \left[\frac{40}{243}\right]^{-1}$$

$$B = \frac{1}{243} \times \frac{243}{40}$$

$$B = \frac{1}{40}$$

**ملاحظة:** تم تعديل الإجابة المتوقعة لـ $B$ إلى $\mathbf{1/40}$ (في الحل أعلاه) بدلاً من 8/3645. سأعدل الإجابة المتوقعة في الكود. (تم التعديل في الكود)

حساب $A$:

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

$$A = -\frac{1}{8} + \frac{9}{4}$$

$$A = -\frac{1}{8} + \frac{18}{8}$$

$$A = \frac{17}{8}$$

حساب $B$:

$$B = (-7)^{(-1) \times (-2)} \times 7^2 + 3^2 \times \frac{5}{3}$$

$$B = (-7)^2 \times 49 + 9 \times \frac{5}{3}$$

$$B = 49 \times 49 + 3 \times 5$$

$$B = 2401 + 15$$

$$B = 2416$$

التصحيح المفصّل للعبارة $C$:

$$C = \frac{1}{5}\sqrt{3} + \frac{1}{10} - \sqrt{17}$$

$$C = \frac{2\sqrt{3}}{10} + \frac{1}{10} - \sqrt{17}$$

$$C = \frac{2\sqrt{3} + 1}{10} - \sqrt{17}$$

$$C = \frac{2\sqrt{3} + 1 - 10\sqrt{17}}{10}$$

التصحيح المفصّل للعبارة $D$:

$$D = (-3\sqrt{3} + 3\sqrt{3}) + \sqrt{7} - 1$$

$$D = 0 + \sqrt{7} - 1$$

$$D = \sqrt{7} - 1$$

التصحيح المفصّل للعبارة $E$:

$$E = \frac{(5+3)\sqrt{5}}{\sqrt{10} \times \frac{1}{2}}$$

$$E = \frac{8\sqrt{5}}{\frac{\sqrt{10}}{2}}$$

$$E = 8\sqrt{5} \times \frac{2}{\sqrt{10}}$$

$$E = \frac{16\sqrt{5}}{\sqrt{10}} = 16\sqrt{\frac{5}{10}}$$

$$E = 16\sqrt{\frac{1}{2}} = \frac{16}{\sqrt{2}}$$

$$E = \frac{16\sqrt{2}}{2} = 8\sqrt{2}$$

التصحيح المفصّل للعبارة $F$:

$$F = \frac{125 \times 0.01}{5\sqrt{2}} \times \frac{1}{10}$$

$$F = \frac{1.25}{5\sqrt{2} \times 10}$$

$$F = \frac{1.25}{50\sqrt{2}}$$

$$F = \frac{1.25\sqrt{2}}{50\times 2} = \frac{1.25\sqrt{2}}{100}$$

$$F = 0.0125\sqrt{2} = \frac{1}{80}\sqrt{2}$$

**ملاحظة:** بالتحويل إلى الصيغة الكسرية التي تم حسابها سابقاً: $0.0125 = 1/80$. الإجابة المتوقعة هي $F = \sqrt{2}/80$. (تم تعديل الإجابة المتوقعة في الكود إلى **sqrt(2)/80**)

التصحيح المفصّل للعبارة:

$$= \left(2+4\sqrt{3}\right) \times \left(2^2 - 2 \times 2\sqrt{3} + (\sqrt{3})^2\right)$$

$$= \left(2+4\sqrt{3}\right) \times \left(4 - 4\sqrt{3} + 3\right)$$

$$= \left(2+4\sqrt{3}\right) \times \left(7 - 4\sqrt{3}\right)$$

$$= 2(7 - 4\sqrt{3}) + 4\sqrt{3}(7 - 4\sqrt{3})$$

$$= 14 - 8\sqrt{3} + 28\sqrt{3} - 16 \times 3$$

$$= 14 + 20\sqrt{3} - 48$$

$$= -34 + 20\sqrt{3}$$

التصحيح المفصّل للعبارة:

**البسط (Numerator):**

$$N = \sqrt{\frac{1}{2}} \times 3\sqrt{3} \times 32 \times \frac{1}{625}$$

$$N = \frac{1}{\sqrt{2}} \times 3\sqrt{3} \times \frac{32}{625}$$

$$N = \frac{\sqrt{2}}{2} \times 3\sqrt{3} \times \frac{32}{625}$$

$$N = \frac{3 \times 16 \sqrt{6}}{625} = \frac{48\sqrt{6}}{625}$$

**المقام (Denominator):**

$$D = 0.5 + 0.05 + 0.005 = 0.555 = \frac{555}{1000} = \frac{111}{200}$$

**النتيجة النهائية (Final Result):**

$$= \frac{N}{D} = \frac{48\sqrt{6}}{625} \times \frac{200}{111}$$

$$= \frac{48\sqrt{6}}{625} \times \frac{200}{111} = \frac{48 \times 8 \sqrt{6}}{25 \times 111} = \frac{384\sqrt{6}}{2775}$$

$$= \frac{128\sqrt{6}}{925}$$ (بعد القسمة على 3)

**الإجابة المتوقعة (للتحقق):** $\mathbf{128*sqrt(6)/925}$

حساب $a$:

$$a = \left(-\frac{2}{3}\right)^2 \times \frac{27}{8}$$

$$a = \frac{4}{9} \times \frac{27}{8}$$

$$a = \frac{4 \times 27}{9 \times 8}$$

$$a = \frac{108}{72}$$

$$a = \frac{3}{2}$$

حساب $b$:

$$b = \left[\frac{1}{4}\right]^{-1} \times \frac{1}{4}$$

$$b = 4 \times \frac{1}{4}$$

$$b = 1$$

حساب $c$:

$$c = \left(\frac{1}{2}\right)^{2\sqrt{7} \times 27 \times 25}$$

$$c = \left(\frac{1}{2}\right)^{2 \times 675 \sqrt{7}}$$

$$c = \left(\frac{1}{2}\right)^{1350\sqrt{7}}$$

حساب $d$:

$$d = \left(\frac{5}{3}\right)^3 \times \frac{8}{5}$$

$$d = \frac{125}{27} \times \frac{8}{5}$$

$$d = \frac{25 \times 8}{27}$$

$$d = \frac{200}{27}$$

حساب $e$:

$$e = \left[2 + \frac{1}{8}\right] \times \frac{1}{4} - \sqrt{4+\sqrt{2}}$$

$$e = \left[\frac{16}{8} + \frac{1}{8}\right] \times \frac{1}{4} - \sqrt{4+\sqrt{2}}$$

$$e = \frac{17}{8} \times \frac{1}{4} - \sqrt{4+\sqrt{2}}$$

$$e = \frac{17}{32} - \sqrt{4+\sqrt{2}}$$

التصحيح المفصّل للمساواة:

**الطرف الأيسر (LHS):**

$$\left(\frac{2}{\sqrt{3}}\right)^5 \times \left(\frac{\sqrt{3}}{2}\right)^{-5} = \left(\frac{2}{\sqrt{3}}\right)^5 \times \left(\frac{2}{\sqrt{3}}\right)^{5}$$

$$= \left(\frac{2}{\sqrt{3}}\right)^{10}$$

**الطرف الأيمن (RHS):**

$$\left(\frac{\sqrt{3}}{2}\right)^8 = \left(\frac{2}{\sqrt{3}}\right)^{-8}$$

**الاستنتاج:** $LHS \neq RHS$. المساواة خاطئة.

**التصحيح:** $\left(\frac{2}{\sqrt{3}}\right)^5 \times \left(\frac{\sqrt{3}}{2}\right)^{-5} = \left(\frac{2}{\sqrt{3}}\right)^{10}$

التصحيح المفصّل للمساواة:

**الطرف الأيسر (LHS):**

$$LHS = \frac{2}{\sqrt{3}} \times \left(\frac{2}{\sqrt{3}}\right)^{2} \times \frac{\sqrt{3}}{2}$$

$$LHS = \left(\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{2}\right) \times \left(\frac{2}{\sqrt{3}}\right)^{2}$$

$$LHS = 1 \times \frac{4}{3} = \frac{4}{3}$$

**الطرف الأيمن (RHS):**

$$RHS = \left(\frac{\sqrt{3}}{2}\right)^3 = \frac{3\sqrt{3}}{8}$$

**الاستنتاج:** $LHS \neq RHS$. المساواة خاطئة.

**التصحيح:** $\frac{2}{\sqrt{3}} \times \left(\frac{\sqrt{3}}{2}\right)^{-2} \times \frac{\sqrt{3}}{2} = \frac{4}{3}$

التصحيح المفصّل للمساواة:

**الطرف الأيسر (LHS):**

$$LHS = \frac{1}{(\sqrt{2})^4} = \frac{1}{4}$$

**الطرف الأيمن (RHS):**

$$RHS = -(8) = -8$$

**الاستنتاج:** $LHS \neq RHS$. المساواة خاطئة.

**التصحيح:** $(\sqrt{2})^{-4} = \frac{1}{4}$

حساب $f$:

$$f = \frac{1}{(\sqrt{3})^4} - \left(-\frac{3}{2}\right)^2$$

$$f = \frac{1}{9} - \frac{9}{4}$$

$$f = \frac{4}{36} - \frac{81}{36}$$

$$f = -\frac{77}{36}$$

حساب $e$:

$$e = \frac{2\sqrt{3}+\sqrt{2}}{3} - \sqrt{5}\frac{\sqrt{3}}{\sqrt{2}}$$

$$e = \frac{2\sqrt{3}+\sqrt{2}}{3} - \frac{\sqrt{15}}{\sqrt{2}}$$

$$e = \frac{2\sqrt{3}+\sqrt{2}}{3} - \frac{\sqrt{30}}{2}$$

$$e = \frac{2(2\sqrt{3}+\sqrt{2})}{6} - \frac{3\sqrt{30}}{6}$$

$$e = \frac{4\sqrt{3} + 2\sqrt{2} - 3\sqrt{30}}{6}$$