a) \(\dfrac{a}{c}=\dfrac{a+b}{c+d}\)

=> a(c + d) = c(a + b)

=> ac + ad = ac + bc

=> ad = bc \(\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\)

b) \(\dfrac{b}{d}=\dfrac{a-b}{c-d}\)

=> b(c - d) = d(a - b)

=> bc - bd = ad - bd

=> bc = ad \(\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\)

a) 2^x(2² - 3) = 32

2^x.1=2⁵

=> x = 5

b) (4x - 3)² = 4x -3

=> (4x - 3)² - (4x - 3) = 0

(4x-3)[(4x - 3) - 1] = 0

(4x-3)(4x - 4)=0

\(\Rightarrow\left[{}\begin{matrix}4x-3=0\\4x-4=0\end{matrix}\right.\)         \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{3}{4}\\x=1\end{matrix}\right.\)

c) 7^2x + 7^2x+3 = 344

=> 7^2x(1 + 7³) =344

7^2x . 344 = 344

=> 2x = 0  => x = 0

d) (7x - 3)²⁰¹² = (3 - 7x)²⁰¹⁰

=> (7x - 3)²⁰¹² - (7x - 3)²⁰¹⁰ = 0

(7x - 3)²⁰¹⁰ [(7x - 3)² - 1] = 0

\(\Rightarrow\left[{}\begin{matrix}7x-3=0\\\left(7x-3\right)^2=1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{3}{7}\\7x=4\\7x=2\end{matrix}\right.\)                 \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{3}{7}\\x=\dfrac{4}{7}\\x=\dfrac{2}{7}\end{matrix}\right.\)

e) (4x² - 3)³ + 8 = 0

(4x² - 3)³ = (-2)³

=> 4x² - 3 = -2

4x² = 1

x² = 1/4

=> \(x=\pm\dfrac{1}{2}\)

a) 13.146 - 46.13 + 4².5 - 6(3² - 4)

= 13(146 - 46) + 4².5 - 6.5

= 13.100 + 5(4² - 6)

= 1300 + 50

=1350

b) 35 + [149 - 2(3³.19 - 3³.17)]

= 35 + {149 - 2[3³(19-17)]}

= 35 + [149 - 2.54]

= 35 + 41 = 76

=> 2^x(2² - 3) = 32

=> 2^x.1=2⁵

=> x = 5

Vậy x=5

\(=2\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{49.51}\right)\)

\(=2\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{47}-\dfrac{1}{49}+\dfrac{1}{49}-\dfrac{1}{51}\right)\)

\(=2\left(1-\dfrac{1}{51}\right)\)

\(=\dfrac{2.50}{51}\)\(=\dfrac{100}{51}\)

\(\dfrac{3}{4}+\dfrac{1}{4}:\left(3x\right)^2=1\)

=> \(\dfrac{1}{4}:9x^2=\dfrac{1}{4}\)

=> 9x² = 1

=> x² = 1/9

=> \(x=\pm\dfrac{1}{3}\)

Vậy ...

P = 2y(2011x + 12x - 2015x)

= 2y.8x

Thay x = -1; y = 2 vào P , ta được:

P = 2.2.8.(-1)

=-32

Vậy P=-32

A = \(\dfrac{3}{17}+\left(\dfrac{2}{3}+\dfrac{3}{17}\right)\)

 = \(\dfrac{3}{17}+\dfrac{43}{51}\)

= \(\dfrac{52}{51}\)

Vậy A = \(\dfrac{52}{51}\)

C = (a + 2)² - (a + 2)(a - 2)

= (a + 2)[a + 2 - (a + 2)]

= 0

D = (3x + 4)² - (x - 4)(x + 4) - 10x

= 9x² + 24x + 16 - (x² - 16) - 10x

= 8x² + 14x + 32