Lời giải:

\(\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)....\left(\frac{1}{100^2}-1\right)\\ =\frac{1-2^2}{2^2}.\frac{1-3^2}{3^2}.\frac{1-4^2}{4^2}....\frac{1-100^2}{100^2}\\ =\frac{(2^2-1)(3^2-1)....(100^2-1)}{2^2.3^2.4^2...100^2}\\ =\frac{(2-1)(2+1)(3-1)(3+1)(4-1)(4+1)....(100-1)(100+1)}{(2.3.4...100)(2.3.4...100)}\)

\(=\frac{(2-1)(3-1)(4-1)...(100-1)}{2.3.4...100}.\frac{(2+1)(3+1)(4+1)...(100+1)}{2.3.4...100}\\ =\frac{1.2.3...99}{2.3.4..100}.\frac{3.4.5..101}{2.3.4...100}\\ =\frac{1}{100}.\frac{101}{2}=\frac{101}{200}\)

Bài 5:

a. Gọi $d=ƯCLN(n-2, n+1)$

$\Rightarrow n-2\vdots d; n+1\vdots d$

$\Rightarrow (n+1)-(n-2)\vdots d$

$\Rightarrow 3\vdots d\Rightarrow d\in \left\{1; 3\right\}$
Để ps tối giản thì $n-2\not\vdots 3$

$\Leftrightarrow n\neq 3k+2$ với $k$ là số tự nhiên bất kỳ.

b.

Gọi $d=ƯCLN(n+5, n-2)$

$\Rightarrow n+5\vdots d; n-2\vdots d$

$\Rightarrow (n+5)-(n-2)\vdots d$

$\Rightarrow 7\vdots d$

$\Rightarrow d\in \left\{1; 7\right\}$

Để ps tối giản thì $n-2\not\vdots 7$

$\Rightarrow n\neq 7k+2$ với $k$ là số tự nhiên bất kỳ.

`x:(-5/3) -(-2/5) =3/10`

`=> x: (-5/3) + 2/5 =3/10`

`=> x:(-5/3) = 3/10-2/5`

`=>x:(-5/3)= 3/10 - 4/10`

`=> x:(-5/3) = -1/10`

`=> x= -1/10 * (-5/3)`

`=> x= 5/30`

`=>x= 1/6`

Vậy `x=1/6`

Lời giải:
$A=\frac{1}{2}+\frac{3}{2}+(\frac{3}{2})^2+(\frac{3}{2})^3+....+(\frac{3}{2})^{2012}$

$\frac{3}{2}A=\frac{3}{4}+(\frac{3}{2})^2+(\frac{3}{2})^3+(\frac{3}{2})^4+...+(\frac{3}{2})^{2013}$

$\Rightarrow \frac{3}{2}A-A=(\frac{3}{2})^{2013}+\frac{3}{4}-\frac{1}{2}-\frac{3}{2}$

$\Rightarrow \frac{1}{2}A=(\frac{3}{2})^{2013}-\frac{5}{4}$

$A=2(\frac{3}{2})^{2013}-\frac{5}{2}$

$\Rightarrow B-A=(\frac{3}{2})^{2013}-2(\frac{3}{2})^{2013}+\frac{5}{2}=\frac{5}{2}-(\frac{3}{2})^{2013}$

\(\dfrac{3}{x-5}=\dfrac{-4}{x-2}\left(x\notin\left\{5;2\right\}\right)\)

\(\Rightarrow3\left(x-2\right)=-4\left(x-5\right)\)

\(\Rightarrow3x-6=-4x+20\)

\(\Rightarrow3x+4x=20+6\)

\(\Rightarrow7x=26\)

\(\Rightarrow x=\dfrac{26}{7}\) (thỏa) 

_________

\(\dfrac{3}{x}=\dfrac{y}{28}=\dfrac{-39}{91}\left(x\ne0\right)\)

\(\Rightarrow\dfrac{3}{x}=\dfrac{y}{28}=\dfrac{-3}{7}\)

+) \(\dfrac{3}{x}=-\dfrac{3}{7}\)

\(\Rightarrow x=\dfrac{3\cdot-7}{3}=-7\) (thỏa) 

+) \(\dfrac{y}{28}=-\dfrac{3}{7}\)

\(\Rightarrow y=\dfrac{28\cdot-3}{7}=-12\)

\(\left(-125\right)\cdot\left(-14\right)\cdot\left(-8\right)\cdot\left(-3\right)\)

\(=\left[\left(-125\right)\cdot\left(-8\right)\right]\cdot\left[\left(-14\right)\cdot\left(-3\right)\right]\)

\(=\left(125\cdot8\right)\cdot\left(14\cdot3\right)\)

\(=1000\cdot42\)

\(=42000\)