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a) \(...=-\dfrac{1}{4}.\dfrac{4}{17}.\left(-\dfrac{63}{21}\right).\left(-\dfrac{7}{12}\right)\)
\(=-\dfrac{1}{17}.\dfrac{63}{21}.\dfrac{7}{12}\)
\(=-\dfrac{7}{68}\)
b) \(...=-\dfrac{2}{5}.\dfrac{4}{15}-\dfrac{3}{10}.\dfrac{4}{15}\)
\(=\dfrac{4}{15}\left(-\dfrac{2}{5}-\dfrac{3}{10}\right)\)
\(=\dfrac{4}{15}\left(-\dfrac{4}{10}-\dfrac{3}{10}\right)\)
\(=\dfrac{4}{15}.\left(-\dfrac{7}{10}\right)=-\dfrac{14}{75}\)
c) \(...=21-\dfrac{15}{4}:\left(\dfrac{9}{24}-\dfrac{4}{24}\right)\)
\(=21-\dfrac{15}{4}:\dfrac{5}{24}\)
\(=21-\dfrac{15}{4}.\dfrac{24}{5}\)
\(=21-3.6=3\)
d) \(...=\left(-\dfrac{3}{4}+\dfrac{2}{5}\right).\dfrac{7}{3}+\left(\dfrac{3}{5}-\dfrac{1}{4}\right).\dfrac{7}{3}\)
\(=\dfrac{7}{3}\left(-\dfrac{3}{4}+\dfrac{2}{5}+\dfrac{3}{5}-\dfrac{1}{4}\right)\)
\(=\dfrac{7}{3}\left(-\dfrac{3}{4}-\dfrac{1}{4}+\dfrac{2}{5}+\dfrac{3}{5}\right)\)
\(=\dfrac{7}{3}\left(-1+1\right)=0\)
1/
$C=5+(5^2+5^3)+(5^4+5^5)+.....+(5^{2022}+5^{2023})$
$=5+5^2(1+5)+5^4(1+5)+....+5^{2022}(1+5)$
$=5+(1+5)(5^2+5^4+....+5^{2022})$
$=5+6(5^2+5^4+....+5^{2022})$
$\Rightarrow C$ chia $6$ dư $5$
$\Rightarrow C\not\vdots 6$
2/
$D=(1+2+2^2)+(2^3+2^4+2^5)+....+(2^{2019}+2^{2020}+2^{2021})$
$=(1+2+2^2)+2^3(1+2+2^2)+....+2^{2019}(1+2+2^2)$
$=(1+2+2^2)(1+2^3+...+2^{2019})$
$=7(1+2^3+...+2^{2019})\vdots 7$
Ta có đpcm.
a: Ư(8)={1;2;4;8}
Ư(12)={1;2;3;4;6;12}
UC(8;12)={1;2;4}
b: B(16)={0;16;32;...}
B(24)={0;24;48;...}
BC(16,24)={0;48;96;...}
6:
Gọi thời gian làm riêng của đội 1 và đội 2 lần lượt là a,b
Trong 1 ngày, đội 1 làm được 1/a(công việc)
Trong 1 ngày, đội 2 làm được 1/b(công việc)
Theo đề, ta có:
1/a+1/b=1/15 và 3/a+5/b=1/4
=>a=24 và b=40
7:
Gọi thời gian chảy riêng đầy bể của vòi 1 và vòi 2 lần lượt là a,b
Theo đề, ta có hệ:
1/a+1/b=1/6 và 3/a+4/b=3/5
=>a=15 và b=10
Bài 1:
a. $-27+(-154)-(-27)+54$
$=(-27)-(-27)+(-154)+54=0-154+54=0-(154-54)=0-100=-100$
b.
$-35.127+(-35).(-27)+700$
$=(-35)(127-27)+700=-35.100+700=-3500+700=-2800$
c.
$-3^4-2[(-2023)^0+(-5)^2]=-81-2(1+25)=-81-2.26=-81-52$
$=-(81+52)=-133$
Bài 2:
a. $-34-2(7-x)=-10$
$2(7-x)=-34-(-10)=-24$
$7-x=-24:2=-12$
$x=7-(-12)=19$
b.
$x=ƯC(36,54,90)$
$\Rightarrow ƯCLN(36,54,90)\vdots x$
$\Rightarrow 18\vdots x$
$\Rightarrow x\in \left\{\pm 1; \pm 2; \pm 3; \pm 6; \pm 9; \pm 18\right\}$
Mà $x>5$ nên $x\in \left\{6; 9; 18\right\}$
3:
\(6a+11b-6\left(a+7b\right)\)
\(=6a+11b-6a-42b=-31b⋮31\)
Ta có: \(\left(6a+11b\right)-6\left(a+7b\right)⋮31\)
\(6a+11b⋮31\)
Do đó: \(6\left(a+7b\right)⋮31\)
=>\(a+7b⋮31\)
Ta có: \(\left(6a+11b\right)-6\left(a+7b\right)⋮31\)
\(a+7b⋮31\)
Do đó: \(6a+11b⋮31\)
4:
\(5a+2b⋮17\)
=>\(12\left(5a+2b\right)⋮17\)
=>\(60a+24b⋮17\)
=>\(51a+17b+9a+7b⋮17\)
=>\(17\left(3a+b\right)+\left(9a+7b\right)⋮17\)
mà \(17\left(3a+b\right)⋮17\)
nên \(9a+7b⋮17\)
m: \(\left(-7x+7\right)\left(2x+100\right)=0\)
=>\(\left[{}\begin{matrix}-7x+7=0\\2x+100=0\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}-7x=-7\\2x=-100\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-50\end{matrix}\right.\)
n: \(2x\left(x+2023\right)\left(-4x+8\right)=0\)
=>\(2\cdot x\left(x+2023\right)\cdot\left(-4\right)\left(x-2\right)=0\)
=>x(x-2)(x+2023)=0
=>\(\left[{}\begin{matrix}x=0\\x-2=0\\x+2023=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\\x=2\\x=-2023\end{matrix}\right.\)
o: \(-2024x\left(4x-4\right)\left(-2x+6\right)=0\)
=>\(x\left(4x-4\right)\left(-2x+6\right)=0\)
=>\(x\cdot4\left(x-1\right)\cdot\left(-2\right)\left(x-3\right)=0\)
=>x(x-1)(x-3)=0
=>\(\left[{}\begin{matrix}x=0\\x-1=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\\x=3\end{matrix}\right.\)
p: -17-2x=199
=>2x=-17-199=-216
=>x=-216/2=-108
q: -24+2x=100
=>2x=100+24=124
=>\(x=\dfrac{124}{2}=62\)
r: \(119-\left(x+5\right)=-21\)
=>\(x+5=119-\left(-21\right)=119+21=140\)
=>x=140-5=135
s: \(-24+\left(-7+x\right)=45\)
=>\(\left(x-7\right)-24=45\)
=>x-31=45
=>x=45+31=76
t: \(-146-\left(-5+x\right)=-6\)
=>\(x-5=-146-\left(-6\right)=-140\)
=>x=-140+5=-135
u: \(-29+4\left(1-5x\right)=-45\)
=>4(1-5x)=-45+29=-16
=>1-5x=-4
=>5x=1+4=5
=>\(x=\dfrac{5}{5}=1\)
v: \(24-5\left(-3+2x\right)=59\)
=>\(24-5\left(2x-3\right)=59\)
=>5(2x-3)=24-59=-35
=>2x-3=-7
=>2x=-4
=>x=-4/2=-2
-17-2x=199
<=>-2x=216
<=>x=-108
-24+2x=100
<=>2x=124
<=>x=62
-24+(-7+x)=45
<=>-7+x=69
<=>x=76
-146-(-5+x)=-6
<=>-146+5-x=-6
<=>x=-135
-29+4(1-5x)=-45
<=>-29+4+20x=-45
<=>20x=-20
<=>x=-1
24-5(-3+2x)=59
<=>24+15-10x=59
<=>-10x=20
<=>x=-2
a: \(\left(-256\right)\cdot45-256\cdot56+256\)
\(=256\left(-45-56+1\right)\)
\(=256\left(-100\right)=-25600\)
b: \(\left(-2\right)^3\cdot1975\cdot\left(-4\right)\cdot\left(-5\right)^3\cdot25\)
\(=\left(-8\right)\cdot\left(-125\right)\cdot\left(-4\right)\cdot25\cdot1975\)
\(=1000\cdot\left(-100\right)\cdot1975=-197500000\)
c: \(2076-1976\cdot65-1976\cdot35\)
\(=2076-1976\left(65+35\right)\)
\(=2076-1976\cdot100=2076-197600=-195524\)
d: \(-437-25\cdot78+25\cdot178\)
\(=-437+25\left(178-78\right)\)
\(=-437+2500=2063\)
a/\(A=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}\)
\(=\dfrac{1}{2}\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{101}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{100}{101}\)
\(=\dfrac{50}{101}\)
b/\(B=\dfrac{4}{1.3}+\dfrac{4}{3.5}+\dfrac{4}{5.7}...+\dfrac{4}{49.51}\)
\(=2\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{49.51}\right)\)
\(=2\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{49}-\dfrac{1}{51}\right)\)
\(=2\left(1-\dfrac{1}{51}\right)\)
\(=2\cdot\dfrac{50}{51}\)
\(=\dfrac{100}{51}\)
c/\(C=\dfrac{6}{3.5}+\dfrac{6}{5.7}+\dfrac{6}{7.9}+...+\dfrac{6}{99.101}\)
\(=3\left(\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+...+\dfrac{2}{99.101}\right)\)
\(=3\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)
\(=3\left(\dfrac{1}{3}-\dfrac{1}{101}\right)\)
\(=3\cdot\dfrac{98}{303}\)
\(=\dfrac{98}{101}\)
d/\(D=\dfrac{1}{3}+\dfrac{1}{15}+\dfrac{1}{35}+...+\dfrac{1}{1023}\)
\(=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{31.33}\)
\(=\dfrac{1}{2}\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{31.33}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{31}-\dfrac{1}{33}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{33}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{32}{33}\)
\(=\dfrac{16}{33}\)
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