1, Ta có :

a . 81 = 3⁴ => 3^x= 3⁴ => x = 4 .

b. 125 = 5³ => 5^x+2 = 5³ =>x + 2 = 3 => x = 1

c. 2³ * 2^{x - 1} = 64

=> 2^{3 + ( x - 1 )} = 64 = 2⁶ 

^{=>  3 + ( x - 1 ) = 6} 

^{=>           x - 1   = 6 - 3 = 3}

                      ^{x  = 3 + 1}

                       ^{x  = 4}

A = 2⁰ . 2¹ . 2² . 2³. 2⁴....2¹⁰⁰

   = 1 . 2¹ . 2² . 2³ . 2⁴ .... 2¹⁰⁰

   = 1 . 2^{1 + 2 + 3 + .... + 100}

Ta có : Số số hạng của dãy số 1 + 2 + 3 + .... + 100 là :

                      (100 - 1) : 1 + 1 = 100 ( số hạng )

Tổng của dãy số 1 + 2 +  3 + ... + 100 là :

                 (100 + 1) . 100 : 2 = 5050

Thay vào, ta được :

A = 1 . 2⁵⁰⁵⁰ = 2⁵⁰⁵⁰

Vậy A = 2⁵⁰⁵⁰

\(A=2^0.2^1.2^2.2^3.....2^{100}=2^1.2^2.2^3......2^{100}=2^{1+2+3+....+100}=2^{\left(1+100\right).\left(100-1+1\right):2}=2^{5050}\)

\(B=6^0.6^1.6^2.6^3.6^4......6^{600}=6^{1+2+3+4+...+600}=6^{\left(1+600\right).\left(600-1+1\right):2}=6^{180300}\)

\(C=7^0.7^1.7^2.7^3.7^4.....7^{700}=7^{0+1+2+3+4+...+700}=7^{\left(700+0\right).\left(700-0+1\right):2}=7^{245000}\)

\(D=8^1.8^2.8^3......8^{800}=8^{1+2+3+....+800}=8^{\left(800+1\right).\left(800-1+1\right):2}=8^{320400}\)

\(A=\frac{15.3^{11}+4.27^1}{9^7}\)

\(\Rightarrow A=\frac{3.5.3^{11}+4.3^{3^1}}{\left(3^2\right)^7}\)

\(\Rightarrow A=\frac{3^{12}.5+4.3^3}{3^{14}}\)

\(\Rightarrow A=\frac{3^3.\left(5.3^8+4.3^3\right)}{3^{14}}\)

\(\Rightarrow A=\frac{32805+4}{177147}\)

\(\Rightarrow A=\frac{32809}{177147}\)

a) \(2^{3x+2}=4^{x+5}\Leftrightarrow2^{3x+2}=2^{2\left(x+5\right)}\Leftrightarrow2^{3x+2}=2^{2x+10}\)

\(\Rightarrow3x+2=2x+10\Leftrightarrow3x+2-2x-10\)

\(\Leftrightarrow x-8=0\Leftrightarrow x=8\) vậy \(x=8\)

câu : b ; c mk cảm thấy đề sai

a: \(=36:4+2\cdot25=9+50=59\)

b: \(=79\left(82+18\right)=79\cdot100=7900\)

c: \(=49-9-\left(4^2+2^2\right)\)

\(=40-16-4=40-20=20\)

d: \(=16+\left[400:\left(200-42-138\right)\right]\)

\(=16+400:20=16+20=36\)

\(19\frac{1}{4}+\frac{1}{2}.2\frac{1}{3}+5,75-\frac{1}{6}+74\)

=\(\frac{77}{4}+\frac{1}{2}.\frac{7}{3}+\frac{575}{100}-\frac{1}{6}+74\)

= \(\frac{77}{4}+\frac{7}{6}+\frac{23}{4}-\frac{1}{6}+74\)

= \(\left(\frac{77}{4}+\frac{23}{4}\right)+\left(\frac{7}{6}-\frac{1}{6}\right)+74\)

= \(\frac{100}{4}+1+74\)

= 100 

^^ 

Đáp án là 100
 

\(S=\frac{3}{\left(1.2\right)^2}+\frac{5}{\left(2.3\right)^2}+...+\frac{61}{\left(30.31\right)^2}\)

\(S=\frac{3}{1^2.2^2}+\frac{5}{2^2.3^2}+...+\frac{61}{30^2.31^2}\)

\(S=\frac{3}{1.4}+\frac{5}{4.9}+...+\frac{61}{900.961}\)

\(S=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{9}+...+\frac{1}{900}-\frac{1}{961}\)

\(S=1-\frac{1}{961}\)

\(S=\frac{960}{961}\)