تمارين تبسيط العبارات الرياضية
الرياضيات - المرحلة الإعدادية
المجموعة الأولى
\[a = (3^5 \times 2^3)^4 \times 3 \times 5^3\]
\[a = (3^5 \times 2^3)^4 \times 3 \times 5^3\]
\[a = 3^{5 \times 4} \times 2^{3 \times 4} \times 3 \times 5^3\]
\[a = 3^{20} \times 2^{12} \times 3^1 \times 5^3\]
\[a = 3^{20+1} \times 2^{12} \times 5^3\]
الناتج النهائي: \[a = 3^{21} \times 2^{12} \times 5^3\]
\[b = 5^4 \times 2^{11} \times 5^3 \times 2 \times 3^{11}\]
\[b = 5^4 \times 2^{11} \times 5^3 \times 2^1 \times 3^{11}\]
\[b = 5^{4+3} \times 2^{11+1} \times 3^{11}\]
\[b = 5^7 \times 2^{12} \times 3^{11}\]
الناتج النهائي: \[b = 5^7 \times 2^{12} \times 3^{11}\]
\[d = (2^4 + 2^4) \times (2^{11} \times 3^4)^2 \times 3\]
\[d = (2^4 + 2^4) \times (2^{11} \times 3^4)^2 \times 3\]
\[d = (2 \times 2^4) \times (2^{11 \times 2} \times 3^{4 \times 2}) \times 3\]
\[d = 2^5 \times 2^{22} \times 3^8 \times 3^1\]
\[d = 2^{5+22} \times 3^{8+1}\]
\[d = 2^{27} \times 3^9\]
الناتج النهائي: \[d = 2^{27} \times 3^9\]
المجموعة الثانية
\[e = (3^{11} + 3^{11} + 3^{11})^2 \times (3^4 \times 2^3)^5\]
\[e = (3^{11} + 3^{11} + 3^{11})^2 \times (3^4 \times 2^3)^5\]
\[e = (3 \times 3^{11})^2 \times (3^{4 \times 5} \times 2^{3 \times 5})\]
\[e = (3^{12})^2 \times 3^{20} \times 2^{15}\]
\[e = 3^{24} \times 3^{20} \times 2^{15}\]
\[e = 3^{44} \times 2^{15}\]
الناتج النهائي: \[e = 3^{44} \times 2^{15}\]
\[f = (3 \times 5^3)^2 \times 3^{11}\]
\[f = (3 \times 5^3)^2 \times 3^{11}\]
\[f = 3^2 \times 5^6 \times 3^{11}\]
\[f = 3^{2+11} \times 5^6\]
\[f = 3^{13} \times 5^6\]
الناتج النهائي: \[f = 3^{13} \times 5^6\]
\[g = (5^3 \times 2)^{11} \times 8000\]
\[g = (5^3 \times 2)^{11} \times 8000\]
\[8000 = 8 \times 1000 = 2^3 \times (2 \times 5)^3 = 2^6 \times 5^3\]
\[g = (5^3 \times 2)^{11} \times 2^6 \times 5^3\]
\[g = 5^{33} \times 2^{11} \times 2^6 \times 5^3\]
\[g = 5^{36} \times 2^{17}\]
الناتج النهائي: \[g = 5^{36} \times 2^{17}\]
المجموعة الثالثة
\[c = 8 \times (11^3 \times 2^4)^2 \times 16^3\]
\[c = 8 \times (11^3 \times 2^4)^2 \times 16^3\]
\[8 = 2^3\] و \[16 = 2^4\]
\[c = 2^3 \times (11^6 \times 2^8) \times 2^{12}\]
\[c = 2^3 \times 11^6 \times 2^8 \times 2^{12}\]
\[c = 11^6 \times 2^{3+8+12}\]
\[c = 11^6 \times 2^{23}\]
الناتج النهائي: \[c = 11^6 \times 2^{23}\]