Đề bài:

\(A = \frac{1 2^{3} \cdot 12 1^{2} \cdot 5 - 2 2^{4} \cdot 3^{3}}{7 5^{2} \cdot 1 1^{4} - 3 0^{2} \cdot 1 1^{5}}\)

Bước 1: Biến đổi các lũy thừa

Ta rút gọn từng số:

• \(1 2^{3} = \left(\right. 3 \cdot 4 \left.\right)^{3} = 3^{3} \cdot 4^{3} = 3^{3} \cdot \left(\right. 2^{2} \left.\right)^{3} = 3^{3} \cdot 2^{6}\)
• \(12 1^{2} = \left(\right. 1 1^{2} \left.\right)^{2} = 1 1^{4}\)
• \(2 2^{4} = \left(\right. 2 \cdot 11 \left.\right)^{4} = 2^{4} \cdot 1 1^{4}\)
• \(3^{3} = 3^{3}\)
• \(7 5^{2} = \left(\right. 3 \cdot 5^{2} \left.\right)^{2} = 3^{2} \cdot 5^{4}\)
• \(1 1^{4} = 1 1^{4}\)
• \(3 0^{2} = \left(\right. 2 \cdot 3 \cdot 5 \left.\right)^{2} = 2^{2} \cdot 3^{2} \cdot 5^{2}\)
• \(1 1^{5} = 1 1^{5}\)

Bước 2: Thay vào biểu thức

Tử số:

\(1 2^{3} \cdot 12 1^{2} \cdot 5 - 2 2^{4} \cdot 3^{3} = \left(\right. 3^{3} \cdot 2^{6} \cdot 1 1^{4} \cdot 5 \left.\right) - \left(\right. 2^{4} \cdot 1 1^{4} \cdot 3^{3} \left.\right)\)

Rút chung:

\(= 3^{3} \cdot 2^{4} \cdot 1 1^{4} \cdot \left(\right. 2^{2} \cdot 5 - 1 \left.\right) = 3^{3} \cdot 2^{4} \cdot 1 1^{4} \cdot \left(\right. 4 \cdot 5 - 1 \left.\right) = 3^{3} \cdot 2^{4} \cdot 1 1^{4} \cdot \left(\right. 20 - 1 \left.\right) = 3^{3} \cdot 2^{4} \cdot 1 1^{4} \cdot 19\)

Mẫu số:

\(7 5^{2} \cdot 1 1^{4} - 3 0^{2} \cdot 1 1^{5} = \left(\right. 3^{2} \cdot 5^{4} \cdot 1 1^{4} \left.\right) - \left(\right. 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 1 1^{5} \left.\right)\)

Rút chung \(3^{2} \cdot 5^{2} \cdot 1 1^{4}\):

\(= 3^{2} \cdot 5^{2} \cdot 1 1^{4} \cdot \left(\right. 5^{2} - 2^{2} \cdot 11 \left.\right) = 3^{2} \cdot 5^{2} \cdot 1 1^{4} \cdot \left(\right. 25 - 4 \cdot 11 \left.\right) = 3^{2} \cdot 5^{2} \cdot 1 1^{4} \cdot \left(\right. 25 - 44 \left.\right) = 3^{2} \cdot 5^{2} \cdot 1 1^{4} \cdot \left(\right. - 19 \left.\right)\)

Bước 3: Viết lại biểu thức đầy đủ

\(A = \frac{3^{3} \cdot 2^{4} \cdot 1 1^{4} \cdot 19}{3^{2} \cdot 5^{2} \cdot 1 1^{4} \cdot \left(\right. - 19 \left.\right)}\)

Rút gọn:

• \(3^{3} / 3^{2} = 3\)
• \(1 1^{4}\) triệt tiêu
• \(19 / \left(\right. - 19 \left.\right) = - 1\)

Còn lại:

\(A = \frac{3 \cdot 2^{4}}{5^{2}} \cdot \left(\right. - 1 \left.\right) = \frac{3 \cdot 16}{25} \cdot \left(\right. - 1 \left.\right) = \frac{48}{25} \cdot \left(\right. - 1 \left.\right) = \boxed{- \frac{48}{25}}\)

✅ Kết quả cuối cùng:

\(\boxed{A = - \frac{48}{25}}\)

Ta có: \(12^3\cdot121^2\cdot5-22^4\cdot3^3\)

\(=\left(2^2\cdot3\right)^3\cdot\left(11^2\right)^2\cdot5-11^4\cdot2^4\cdot3^3\)

\(=2^6\cdot3^3\cdot11^4\cdot5-11^4\cdot2^4\cdot3^3=11^4\cdot3^3\cdot2^4\left(2^2\cdot5-1\right)\)

\(=11^4\cdot3^3\cdot2^4\cdot19\)

Ta có: \(75^2\cdot11^4-30^2\cdot11^5\)

\(=\left(3\cdot5^2\right)^2\cdot11^4-\left(2\cdot3\cdot5\right)^2\cdot11^5\)

\(=3^2\cdot5^4\cdot11^4-2^2\cdot3^2\cdot5^2\cdot11^5\)

\(=3^2\cdot5^2\cdot11^4\left(5^2-2^2\cdot11\right)=3^2\cdot5^2\cdot11^4\cdot\left(-19\right)\)

Ta có: \(A=\frac{12^3\cdot121^2\cdot5-22^4\cdot3^3}{75^2\cdot11^4-30^2\cdot11^5}\)

\(=\frac{2^4\cdot3^3\cdot11^4\cdot19}{3^2\cdot5^2\cdot11^4\cdot\left(-19\right)}=\frac{2^4\cdot3}{5^2\cdot\left(-1\right)}=\frac{48}{-25}\)