1. TÍNH
a)

Tính:

5,9⋅106−4,8⋅3,8⋅10815,2⋅10−6−7,29⋅109\frac{5{,}9 \cdot 10^6 - 4{,}8 \cdot 3{,}8 \cdot 10^8}{15{,}2 \cdot 10^{-6} - 7{,}29 \cdot 10^9}15,2⋅10−6−7,29⋅1095,9⋅106−4,8⋅3,8⋅108​

Ta có:

4,8⋅3,8=18,24⇒Tửso^ˊ=5,9⋅106−18,24⋅1084{,}8 \cdot 3{,}8 = 18{,}24 \Rightarrow Tử số = 5{,}9 \cdot 10^6 - 18{,}24 \cdot 10^84,8⋅3,8=18,24⇒Tửso^ˊ=5,9⋅106−18,24⋅108 =0,0059⋅109−1,824⋅109=−1,8181⋅109= 0{,}0059 \cdot 10^9 - 1{,}824 \cdot 10^9 = -1{,}8181 \cdot 10^9=0,0059⋅109−1,824⋅109=−1,8181⋅109

Mẫu số:

15,2⋅10−6−7,29⋅109≈−7,29⋅10915{,}2 \cdot 10^{-6} - 7{,}29 \cdot 10^9 \approx -7{,}29 \cdot 10^915,2⋅10−6−7,29⋅109≈−7,29⋅109

Vậy:

−1,8181⋅109−7,29⋅109=1,81817,29≈0,2494\frac{-1{,}8181 \cdot 10^9}{-7{,}29 \cdot 10^9} = \frac{1{,}8181}{7{,}29} \approx 0{,}2494−7,29⋅109−1,8181⋅109​=7,291,8181​≈0,2494

Đáp số: 0,2494

b)

Tính:

12+5878=12+2939⇒12⋅39+2939=468+2939=4973912 + \frac{58}{78} = 12 + \frac{29}{39} \Rightarrow \frac{12 \cdot 39 + 29}{39} = \frac{468 + 29}{39} = \frac{497}{39}12+7858​=12+3929​⇒3912⋅39+29​=39468+29​=39497​

Đáp số: 49739\frac{497}{39}39497​

c)

Tính:

712+720+790=2222712 + 720 + 790 = 2222712+720+790=2222

Đáp số: 2222

d)

Tính:

A=−7+72+73+74+2019=−7+49+343+2401+2019=4848−7=4841A = -7 + 7^2 + 7^3 + 7^4 + 2019 = -7 + 49 + 343 + 2401 + 2019 = 4848 - 7 = 4841A=−7+72+73+74+2019=−7+49+343+2401+2019=4848−7=4841

Đáp số: 4841

e)

Tính:

15+3+1213=18+1213=234+1213=2461315 + 3 + \frac{12}{13} = 18 + \frac{12}{13} = \frac{234 + 12}{13} = \frac{246}{13}15+3+1312​=18+1312​=13234+12​=13246​

Đáp số: 24613\frac{246}{13}13246​

2. GIẢI PHƯƠNG TRÌNH / HỆ PHƯƠNG TRÌNH
a)
x−23+x−24+x−25=0⇒3x−72=0⇒x=723=24x - 23 + x - 24 + x - 25 = 0 \Rightarrow 3x - 72 = 0 \Rightarrow x = \frac{72}{3} = 24x−23+x−24+x−25=0⇒3x−72=0⇒x=372​=24

Đáp số: x = 24

b)
x2+x+1=−2⇒x2+x+3=0⇒Δ=12−4⋅1⋅3=−11<0x^2 + x + 1 = -2 \Rightarrow x^2 + x + 3 = 0 \Rightarrow \Delta = 1^2 - 4 \cdot 1 \cdot 3 = -11 < 0x2+x+1=−2⇒x2+x+3=0⇒Δ=12−4⋅1⋅3=−11<0

Phương trình vô nghiệm.

c)
x−84+2x−72−x−64+x−60=10⇒3x−280=10⇒3x=290⇒x=2903x - 84 + 2x - 72 - x - 64 + x - 60 = 10 \Rightarrow 3x - 280 = 10 \Rightarrow 3x = 290 \Rightarrow x = \frac{290}{3}x−84+2x−72−x−64+x−60=10⇒3x−280=10⇒3x=290⇒x=3290​

Đáp số: x=2903x = \frac{290}{3}x=3290​

d)
(x+3)2+(y−2)2=0⇒{x+3=0⇒x=−3y−2=0⇒y=2(x + 3)^2 + (y - 2)^2 = 0 \Rightarrow \begin{cases} x + 3 = 0 \Rightarrow x = -3 \\ y - 2 = 0 \Rightarrow y = 2 \end{cases}(x+3)2+(y−2)2=0⇒{x+3=0⇒x=−3y−2=0⇒y=2​

Đáp số: x = -3, y = 2

e)
(2x+3)(y−2)=6,x∈Z(2x + 3)(y - 2) = 6,\quad x \in \mathbb{Z}(2x+3)(y−2)=6,x∈Z

Gọi a=2x+3⇒a∈Z,a∣6⇒a∈{±1,±2,±3,±6}a = 2x + 3 \Rightarrow a \in \mathbb{Z}, a \mid 6 \Rightarrow a \in \{\pm1, \pm2, \pm3, \pm6\}a=2x+3⇒a∈Z,a∣6⇒a∈{±1,±2,±3,±6}

Tìm x:

2x+3=1⇒x=−12x+3=−1⇒x=−22x+3=3⇒x=02x+3=−3⇒x=−3\begin{aligned} &2x + 3 = 1 \Rightarrow x = -1 \\ &2x + 3 = -1 \Rightarrow x = -2 \\ &2x + 3 = 3 \Rightarrow x = 0 \\ &2x + 3 = -3 \Rightarrow x = -3 \\ \end{aligned}​2x+3=1⇒x=−12x+3=−1⇒x=−22x+3=3⇒x=02x+3=−3⇒x=−3​

Đáp số: x∈{−3,−2,−1,0}x \in \{ -3, -2, -1, 0 \}x∈{−3,−2,−1,0}

3. KẾT LUẬN

Xét biểu thức:

A=x−7x+1∈Z,x≠−1A = \frac{x - 7}{x + 1} \in \mathbb{Z}, \quad x \ne -1A=x+1x−7​∈Z,x=−1

Điều kiện để A nguyên:

x+1∣x−7⇒x+1∣(x−7−x−1)=−8⇒x+1∈{±1,±2,±4,±8}⇒x∈{0,−2,1,−3,3,−9}x + 1 \mid x - 7 \Rightarrow x + 1 \mid (x - 7 - x - 1) = -8 \Rightarrow x + 1 \in \{ \pm1, \pm2, \pm4, \pm8 \} \Rightarrow x \in \{ 0, -2, 1, -3, 3, -9 \}x+1∣x−7⇒x+1∣(x−7−x−1)=−8⇒x+1∈{±1,±2,±4,±8}⇒x∈{0,−2,1,−3,3,−9}

Loại x=−1x = -1x=−1 (vì mẫu bằng 0)

Kết luận: x∈{−9,−3,−2,0,1,3}x \in \{ -9, -3, -2, 0, 1, 3 \}x∈{−9,−3,−2,0,1,3}