(x-1)²⁰²⁰=(x-1)²⁰²²

=>(x-1)²⁰²⁰-(x-1)²⁰²²=0

=>(x-1)²⁰²⁰-(x-1)²⁰²⁰.(x-1)²=0

=>(x-1)²⁰²⁰(1-(x-1)²=0

=>(x-1)²⁰²⁰=0 hoặc 1-(x-1)²=0

=>x=1 hoặc x=2.

Bài 2

a,2¹⁰⁵ và 5⁴⁵

2¹⁰⁵=(2⁷)¹⁵=128¹⁵

5⁴⁵=(5³)¹⁵=125¹⁵

Vì 128^15_>12515 nên 2¹⁰⁵>5⁴⁵.

b,

5⁵⁴ và 3⁸¹

5⁵⁴=(5⁶)⁹=15625⁹

3⁸¹=(3⁹)⁹=19683⁹

Vì 15625⁹<19683⁹ nên 5⁵⁴<3⁸¹

Bài 1 :

\(\left(x-1\right)^{2020}=\left(x-1\right)^{2022}\)

\(\Rightarrow\left(x-1\right)^{2022}-\left(x-1\right)^{2020}=0\)

\(\Rightarrow\left(x-1\right)^{2020}\left[\left(x-1\right)^2-1\right]=0\)

\(\Rightarrow\left[{}\begin{matrix}x-1=0\\\left(x-1\right)^2-1=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\\left(x-1\right)^2=1\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x-1=1\\x-1=-1\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x=2\\x=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x=2\end{matrix}\right.\)

2^161 > 2^160 = ( 2^4 )^40 = 16^40

Vì : 16^40 > 13^40 Suy ra 13^40 < 2^161

Vậy : 13^40 < 2^161

Ta sử dụng tính chất bắt cầu :

Ta thấy : \(2^{161}>2^{160}\)

Mà \(2^{160}=\left(2^4\right)^{40}=16^{40}\)

Ta so sánh :

\(16^{40}>13^{40}\Rightarrow13^{40}< 2^{161}\)

13⁴⁰ và 2¹⁶¹

Ta có : 

2¹⁶¹ > 2¹⁶⁰ = ( 2⁴ )⁴⁰ = 16⁴⁰

Vì 13⁴⁰ < 16⁴⁰ 

Nên 13⁴⁰ < 2¹⁶¹

a) Ta có:

\(2^{300}=\left(2^3\right)^{100}=8^{100}\)

\(3^{200}=\left(3^2\right)^{100}=9^{100}\)

Vì \(8^{100}< 9^{100}\) nên \(2^{300}< 3^{200}\)

b) Ta có:

\(2^{161}>2^{160}=\left(2^4\right)^{40}=16^{40}\)

Vì \(16^{40}>13^{40}\) nên \(13^{40}< 2^{161}\)

Ta có :

\(2^{161}=\left(2^4\right)^{40}.2=16^{40}.2\)

Vì \(16^{40}.2>13^{40}\)

\(\Rightarrow2^{161}>13^{40}\)

Ta có: \(2^{161}>2^{160}=\left(2^4\right)^{40}=16^{40}\)

Vì \(16^{40}>13^{40}\) nên \(2^{160}>13^{40}\Rightarrow2^{161}>13^{40}\)

Ta có : 125⁸⁰ = ( 5³)⁸⁰ = 5²⁴⁰    và 25¹¹⁸ = ( 5²)¹¹⁸ = 5²³⁶

Mà 5²⁴⁰   < 5²³⁶ nên 125⁸⁰ < 25¹¹⁸