45¹⁰.5³⁰=3²⁰.5¹⁰.5³⁰=3²⁰.5⁴⁰=3²⁰.(5²)²⁰=75²⁰

->điều phải chứng minh.

\(75^{20}=\left(5^2\right)^{20}.3^{20}=5^{40}.3^{20}\); \(45^{10}.5^{30}=\left(3^2\right)^{10}.5^{10}.5^{30}=3^{20}.5^{40}\)

Vậy \(75^{20}=45^{10}.5^{30}\left(=5^{40}.3^{20}\right)\)

bạn viết có dấu vào bài đi, khó hiểu quá

12⁸.9¹²

=(2².3)⁸.9¹²

=(2²)⁸.3⁸.9¹²

=2¹⁶.(3²)⁴.9¹²

=2¹⁶.9⁴.9¹²

=2¹⁶.9¹⁶

=(2.9)¹⁶

=18¹⁶

12⁸.9¹²=3⁸.4⁸.9¹²=9⁴.4⁸.9¹²=9¹⁶.2¹⁶=18¹⁶

\(\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right).....\left(1+\frac{1}{99.101}\right)\)

\(=\frac{2.2}{1.3}\frac{3.3}{2.4}.....\frac{100.100}{99.101}\)

\(=\frac{\left(2.3.4.....100\right).\left(2.3.4.....100\right)}{\left(1.2.3.....99\right).\left(3.4.5.....101\right)}\)

\(=\frac{100.2}{101}=\frac{200}{101}\)

\(\frac{\left(x-3\right)\left(x+5\right)}{\left(x-2\right)^2}< 0\)

\(\Rightarrow\frac{\left(x-3\right)\left(x+5\right)}{\left(x-2\right).\left(x-2\right)}< 0\)

=> ( x - 3 ) . ( x - 5 ) và ( x - 2 ) . ( x - 2 ) trái dấu 

Mà ( x - 2 )² = ( x - 2 ) . ( x - 2 ) ≥ 0 ∀ x

 \(\Rightarrow\hept{\begin{cases}\left(x−3\right).\left(x+5\right)< 0\\\left(x−2\right).\left(x−2\right)>0\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}x< −5;−5< x< 3\\x>2\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}x< −5\\2< x< 3\end{cases}}\)