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\(a^3b-ab^3=ab\left(a^2-b^2\right)=ab\left(a^2-ab+ab-b^2\right)=ab\left(a-b\right)\left(a+b\right)\)
Với a hoặc b chẵn \(\Leftrightarrow ab\left(a-b\right)\left(a+b\right)⋮2\)
Với a và b lẻ \(\Leftrightarrow\left(a-b\right)⋮2\Leftrightarrow ab\left(a-b\right)\left(a+b\right)⋮2\)
Vậy \(ab\left(a-b\right)\left(a+b\right)⋮2,\forall a,b\left(1\right)\)
Với a hoặc b chia hết cho 3 thì \(ab\left(a-b\right)\left(a+b\right)⋮3\)
Với \(a=3k+1;b=3q+1\Leftrightarrow\left(a-b\right)=3\left(k-q\right)⋮3\)
\(\Leftrightarrow ab\left(a-b\right)\left(a+b\right)⋮3\)
Với \(a=3k+1;b=3q+2\Leftrightarrow\left(a+b\right)=\left(3k+1+3q+2\right)=3\left(k+q+1\right)⋮3\)
\(\Leftrightarrow ab\left(a-b\right)\left(a+b\right)⋮3\)
Mà a,b có vai trò tương đương nên \(ab\left(a-b\right)\left(a+b\right)⋮3,\forall a,b\left(2\right)\)
\(\left(1\right)\left(2\right)\Leftrightarrowđpcm\)
Ta có : a3b -ab3
=a3b -ab -ab3 +ab
=ab (a2 -1) -ab (b2 -1)
=ab (a-1)(a+1) -ab (b-1)(b+1)
Vì a (a-1)(a+1) là 3 số tự nhiên liên tiếp nên chia hết cho 6 .Tương tự b (b-1)(b+1) cũng chia hết cho 6
=> a3b -ab3 chia hết cho 6 (đpcm )
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
3: \(\Leftrightarrow x\cdot\dfrac{-3}{5}=\dfrac{1}{4}+\dfrac{3}{4}=1\)
nên x=-5/3
4: \(\Leftrightarrow x\cdot\dfrac{4}{28}=\dfrac{28}{3}\cdot\dfrac{3}{28}=1\)
hay x=28/4=7
5: \(\Leftrightarrow x:\left(\dfrac{7}{2}-\dfrac{31}{6}\right)=\dfrac{21}{5}-\dfrac{20}{3}\)
\(\Leftrightarrow x:\dfrac{-5}{3}=\dfrac{-37}{15}\)
\(\Leftrightarrow x=\dfrac{37}{15}\cdot\dfrac{5}{3}=\dfrac{37}{9}\)
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65 - 5 . ( x + 2 ) = 15
5 . ( x + 2 ) = 65 - 15
5 . ( x + 2 ) = 50
( x + 2 ) = 50 : 5
x + 2 = 10
x = 8
Vậy x = 8
\(65-5.\left(x+2\right)=15\)
\(5\left(x+2\right)=50\)
\(x+2=10\)
\(x=8\)
![](https://rs.olm.vn/images/avt/0.png?1311)
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3:
a: =8/24+9/24-14/24=3/24=1/8
b: =-12/56+35/56-28/56=-5/56
c: =9/36-24/36-22/36=-37/36
d: \(=\dfrac{6}{24}+\dfrac{10}{24}-\dfrac{21}{24}-\dfrac{1}{13}=\dfrac{-5}{24}-\dfrac{1}{13}=\dfrac{-89}{24\cdot13}\)
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\(A=\dfrac{1}{2}+\dfrac{2}{4}+\dfrac{3}{8}+...+\dfrac{10}{2^{10}}\)
\(2A=\dfrac{1}{1}+\dfrac{2}{2}+\dfrac{3}{4}+...+\dfrac{10}{2^9}\)
\(2A-A=\left(1+\dfrac{2}{2}+\dfrac{3}{4}+...+\dfrac{10}{2^9}\right)-\left(\dfrac{1}{2}+\dfrac{2}{4}+...+\dfrac{10}{2^{10}}\right)\)
\(A=1+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2^9}-\dfrac{10}{2^{10}}\)
\(B=1+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2^9}\)
\(2B=2+1+\dfrac{1}{2}+...+\dfrac{1}{2^8}\)
\(2B-B=\left(2+1+\dfrac{1}{2}+...+\dfrac{1}{2^8}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2^9}\right)\)
\(B=2-\dfrac{1}{2^9}\)
Suy ra \(A=B-\dfrac{10}{2^{10}}=2-\dfrac{1}{2^9}-\dfrac{10}{2^{10}}=\dfrac{509}{256}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
a) \(7^2-7\left(13-x\right)=14\)
\(7\left(13-x\right)=49-14=35\)
\(13-x=5\)
\(x=13-5=8\)
b) \(5x-5^2=10\)
\(5x=10+25=35\)
\(x=7\)
c) \(4\left(x-5\right)-2^3=2^4.3=48\)
\(4\left(x-5\right)=48+8=56\)
\(x-5=14\)
\(x=19\)
4:
a: \(\Leftrightarrow49+7\left(x-13\right)=14\)
=>7(x-13)=35
=>x-13=5
=>x=18
b: \(5x-5^2=10\)
=>\(5x=10+25=35\)
=>x=7
c: \(4\left(x-5\right)-2^3=2^4\cdot3\)
=>\(4\left(x-5\right)=16\cdot3+8=56\)
=>x-5=14
=>x=19
Bài 1:
a, 19 + (-45)
= - (45 - 19)
= - 26
b, (-57) + 12
= - (57 - 12)
= - 45
c, 80 + (-17)
= 80 - 17
= 63
d, (-13) + (-52)
= - (13 + 52)
= - 65
Bài 2: - 37; 54; -70; -163; 246; 12
Các số trên được sắp xếp theo thứ tự tăng dần là:
-163; - 70; -37; 2; 12; 54; 246