\(\dfrac{12^4.\left(-10^2\right)}{3^4.4^5.5^2}=\dfrac{3^4.2^8.2^2.\left(-5^2\right)}{3^4.2^{10}.5^2}=\dfrac{3^4.2^{10}.\left(-5^2\right)}{3^4.2^{10}.5^2}=-1\)

1.1!+2.2!+3.3!+4.4!+5.5! 

= (2-1).1! + (3-1).2! + (4-1).3! + (5-1).4! + (6-1).5! 

= 2! - 1! + 3! - 2! + 4!-3! + 5! - 4! + 6! - 5! 

= 6! - 1!

= 720-1

= 719

1!+2.2!+3.3!+4.4!+5.5! 
=(2-1).1!+(3-1).2!+(4-1).3!+ 
(5-1).4!+(6-1).5! 
=2!-1!+3!-2!+4!-3!+5!-4!+6!-5! 
=6!-1!=720-1=719 

\(=\dfrac{2^4\cdot5^4+2^5\cdot5^3}{2^{10}\cdot16}=\dfrac{2^4\cdot5^3\left(5+2\right)}{2^{10}\cdot2^4}=\dfrac{2^4\cdot5^3\cdot7}{2^{14}}=\dfrac{5^3\cdot7}{2^{10}}=\dfrac{875}{1024}\)

\(\left(-\frac{10}{3}\right)^5.\left(-\frac{6}{5}\right)^4\)

\(=\frac{-2^5.5^5}{3^5}.\frac{2^4.3^4}{5^4}\)

\(=\frac{-2^9.5}{3}=-\frac{2560}{3}\)

• \(\frac{4^6.3^4.9^5}{6^{12}}=\frac{\left(2^2\right)^6.3^4.\left(3^2\right)^5}{\left(2.3\right)^{12}}=\frac{2^{12}.3^4.3^{10}}{2^{12}.3^{12}}=\frac{2^{12}.3^{14}}{2^{12}.3^{12}}=3^2=9\)
• ​​\(\frac{3^{10}.11+9^5.5}{3^9.2^4}=\frac{3^{10}.11+\left(3^2\right)^5.5}{3^9.16}=\frac{3^{10}.11+3^{10}.5}{3^9.16}=\frac{3^{10}.\left(11+5\right)}{3^9.16}=\frac{3^{10}.16}{3^9.16}=3\)
• 2¹⁰⁰ - 2⁹⁹ - 2⁹⁸ - ... - 2² - 2

= 2¹⁰⁰ - (2⁹⁹ + 2⁹⁸ + ... + 2² + 2)

Đặt A = 2⁹⁹ + 2⁹⁸ + ... + 2² + 2

2A = 2¹⁰⁰ + 2⁹⁹ + ... + 2³ + 2²

2A - A = (2¹⁰⁰ + 2⁹⁹ + ... + 2³ + 2²) - (2⁹⁹ + 2⁹⁸ + ... + 2² + 2)

A = 2¹⁰⁰ - 2

Ta có:

2¹⁰⁰ - 2⁹⁹ - 2⁹⁸ - ... - 2² - 2

= 2¹⁰⁰ - (2¹⁰⁰ - 2)

= 2¹⁰⁰ - 2¹⁰⁰ + 2

= 0 + 2

= 2

• 3⁸ : 3⁶ + (2²)⁴ : 2⁹

= 3² + 2⁸ : 2⁹

\(=9+\frac{1}{2}\)

\(=\frac{18}{2}+\frac{1}{2}=\frac{19}{2}\)