Ta có :

( -2 )¹⁰⁰ = [( -2 )⁴ ]²⁵ = 16²⁵ (1)

3⁷⁵ = ( 3³ )²⁵ = 27²⁵ (2)

5⁵⁰ = ( 5² )²⁵ = 25²⁵ (3)

Từ (1) ; (2) và (3) => ( -2 )¹⁰⁰ < 5⁵⁰ < 3⁷⁵

Giải thích các bước giải:

 a)2^100=(2^4)^25=16^25;3^75=(3^3)^25=9^25;5^50=25^25

   vì 9^25<16^25<25^25=>3^75<2^100<5^50

b)10^30=(10^3)^10=1000^10

     mà  8^10<1000^10

   =>10^30>8^10

c)3^40=81^10;37^20=1369^10

 vì 15^10<81^10<1369^10

   =>15^10<3^40<37^20

Ta có :

| 1 - x | + | 3 - x | = | x - 1 | + | 3 - x | ≥ | x - 1 + 3 - x | = | 2 | = 2

=> | 1 - x | + | 3 - x | + 2015 ≥ 2017

Dấu "=" xảy ra <=> \(\left(x-1\right)\left(3-x\right)=0\)

<=> \(\hept{\begin{cases}1\le x\\x\le3\end{cases}}\Leftrightarrow1\le x\le3\)

Ta có :

\(A=\frac{c}{a_1.a_2}+\frac{c}{a_2.a_3}+...+\frac{c}{a_{n-1}.a_n}\)

\(\Rightarrow\frac{1}{c}A=\frac{1}{a_1.a_2}+\frac{1}{a_2.a_3}+...+\frac{1}{a_{n-1}.a_n}\)

\(\Rightarrow\frac{k}{c}A=\frac{k}{a_1.a_2}+\frac{k}{a_2.a_3}+...+\frac{k}{a_{n-1}.a_n}\)

\(\Rightarrow\frac{k}{c}A=\frac{a_2-a_1}{a_1.a_2}+\frac{a_3-a_2}{a_2.a_3}+...+\frac{a_n-a_{n-1}}{a_{n-1}.a_n}\)

\(\Rightarrow\frac{k}{c}A=\frac{1}{a_1}-\frac{1}{a_2}+\frac{1}{a_2}-\frac{1}{a_3}+...+\frac{1}{a_{n-1}}-\frac{1}{a_n}\)

\(\Rightarrow\frac{k}{c}A=\frac{1}{a_1}-\frac{1}{a_n}\)\(\Rightarrow A=\left(\frac{1}{a_1}-\frac{1}{a_n}\right):\frac{k}{c}\)

a) Ta có :

O = A₄ (vì là 2 góc so le trong)

=> a // Ox

b) Ta có :

B₁ = A₃ (vì là 2 góc so le trong)

=> b // Oy

\(\frac{kA}{c}=\frac{k}{a_1.a_2}+\frac{k}{a_2.a_3}+\frac{k}{a_3.a_4}+...+\frac{k}{a_{n-1}.a_n}=\)

\(=\frac{a_2-a_1}{a_1.a_2}+\frac{a_3-a_2}{a_{2.}.a_3}+\frac{a_4-a_3}{a_3.a_4}+..+\frac{a_n-a_{n-1}}{a_{n-1}.a_n}=\)

\(=\frac{1}{a_1}-\frac{1}{a_2}+\frac{1}{a_2}-\frac{1}{a_3}+\frac{1}{a_3}-\frac{1}{a_4}+...+\frac{1}{a_{n-1}}-\frac{1}{a_n}=\)

\(=\frac{1}{a_1}-\frac{1}{a_n}=\frac{a_n-a_1}{a_1.a_n}\Rightarrow A=\frac{\left(a_n-a_1\right).c}{a_1.a_n.k}\)

Ta có\(\frac{1+4y}{13}=\frac{1+6y}{19}=\frac{1+8y}{5x}=\frac{1+4y+1+8y}{13+5x}=\frac{2\left(1+6y\right)}{13+5x}\)

=> \(\frac{1+6y}{19}=\frac{2\left(1+6y\right)}{13+5x}\)

=> 2(1 + 6y).19 = (13 + 5x).(1 + 6y)

=> 38 = 13 + 5x

<=> 25 = 5x

<=> x = 5

Khi đó \(\frac{1+4y}{13}=\frac{1+6y}{19}\)

<=> (1 + 4y).19 = (1 + 6y).13

<=> 19 + 76y = 13 + 78y 

<=> 6 = 2y

<=> y = 3

Vậy y = 3 ; x = 5