\(\frac{a-1}{2}=\frac{b-2}{3}=\frac{c-3}{4}\)và a+b-c=9
Những câu hỏi liên quan
\(A=\frac{11}{9}-\frac{7}{8}+\frac{-2}{3}-\frac{1}{8}+\frac{25}{9}-\frac{4}{3}\)
\(B=1\frac{3}{4}:\frac{3}{5}-\frac{2}{3}.1,75+\left(\frac{1}{2}\right)^2:\frac{1}{7}\)
a) Tính A và B
b) Tìm C biết (A-2.B) của C bằng 6
giúp mình giải nha
\(A=\frac{11}{9}-\frac{7}{8}+-\frac{2}{3}-\frac{1}{8}+\frac{25}{9}-\frac{4}{3}\)
\(A=1\)
\(B=1\frac{3}{4}:\frac{3}{5}-\frac{2}{3}x1,75+\left(\frac{1}{2}\right)^2:\frac{1}{7}\)
\(B=3,5\)
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C/m các BĐT sau :
1.a^3-3a+4ge b^3-3b
2,frac{1}{frac{1}{a+c}+frac{1}{b+d}}gefrac{1}{frac{1}{a}+frac{1}{b}}+frac{1}{frac{1}{c}+frac{1}{d}} với a, b, c, d0
3,a^3+b^3gefrac{1}{4};a+bge1
4, a^3+b^3le a^4+b^4;a+bge2
5, left(a+bright)left(a^3+b^3right)left(a^5+b^5right)le4left(a^9+b^9right);a,bge0
6, frac{c+a}{sqrt{a^2+c^2}}gefrac{c+b}{sqrt...
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C/m các BĐT sau :
\(1.a^3-3a+4\ge b^3-3b
\)
\(2,\frac{1}{\frac{1}{a+c}+\frac{1}{b+d}}\ge\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}}\) với a, b, c, d>0
\(3,a^3+b^3\ge\frac{1}{4};a+b\ge1\)
4, \(a^3+b^3\le a^4+b^4;a+b\ge2\)
5, \(\left(a+b\right)\left(a^3+b^3\right)\left(a^5+b^5\right)\le4\left(a^9+b^9\right);a,b\ge0\)
6, \(\frac{c+a}{\sqrt{a^2+c^2}}\ge\frac{c+b}{\sqrt{c^2+b^2}};a>b>0,c>\sqrt{ab}\)
Các bn làm đc bài nào thì giúp mk với, cảm ơn ạ !
1.
C/m bổ đề: \(a^3-b^3\ge\frac{1}{4}\left(a^3-b^3\right)\) với \(\forall a,b\in R,a\ge b\)
\(\Leftrightarrow4a^3-4b^3-\left(a^3-3a^2b+3ab^2-b^3\right)\ge0\)
\(\Leftrightarrow3a^3+3a^2b-3ab^2-3b^3\ge0\)
\(\Leftrightarrow3\left(a^2-b^2\right)\left(a+b\right)\ge0\)
\(\Leftrightarrow3\left(a+b\right)^2\left(a-b\right)\ge0\)(đúng)
Theo bài ra: \(a^3-b^3\ge3a-3b-4\)
\(\Leftrightarrow\) Cần c/m: \(\left(a-b\right)^3\ge12a-12b-16\)(1)
Thật vậy:
\(\left(1\right)\)\(\Leftrightarrow\left(a-b\right)^3-12\left(a-b\right)+16\ge0\)
\(\Leftrightarrow\left[\left(a-b\right)^3-8\right]-12\left(a-b-2\right)\ge0\)
\(\Leftrightarrow\left(a-b-2\right)\left[\left(a-b\right)^2+2\left(a-b\right)+4\right]-12\left(a-b-2\right)\ge0\)
\(\Leftrightarrow\left(a-b-2\right)\left[\left(a-b\right)^2+2\left(a+b\right)-8\right]\ge0\)
\(\Leftrightarrow\left(a-b-2\right)^2\left(a-b+4\right)\ge0\) (đúng với mọi a,b thỏa mãn \(a,b\in R,a\ge b\))
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2.
\(BĐT\Leftrightarrow\frac{1}{\frac{a+b}{ab}}+\frac{1}{\frac{c+d}{cd}}\le\frac{1}{\frac{a+b+c+d}{\left(a+c\right)\left(b+d\right)}}\)
\(\Leftrightarrow\frac{ab}{a+b}+\frac{cd}{c+d}\le\frac{\left(a+c\right)\left(b+d\right)}{a+b+c+d}\)
\(\Leftrightarrow\frac{ab\left(c+d\right)+cd\left(a+b\right)}{\left(a+b\right)\left(c+d\right)}\le\)\(\frac{ab+ad+bc+cd}{a+b+c+d}\)
\(\Leftrightarrow\frac{abc+abd+acd+bcd}{ac+ad+bc+bd}\le\frac{ab+ad+bc+cd}{a+b+c+d}\)
\(\Leftrightarrow\left(ad+ab+bc+cd\right)\left(ac+ad+bc+bd\right)\ge\)\(\left(a+b+c+d\right)\left(abc+abd+acd+bcd\right)\)
\(\Leftrightarrow\left(ad\right)^2-2abcd+\left(bc\right)^2\ge0\)
\(\Leftrightarrow\left(ad-bc\right)^2\ge0\) (đúng với mọi a,b,c,d>0)
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Ôn tập Bất đẳng thức
1 , Cho a,b,c3 thỏa mãn abc(a+b+c)3 . Tìm GTNN của C frac{a}{sqrt{9-b^2}}+frac{b}{sqrt{9-c^2}}+frac{c}{sqrt{9-a^2}}
2, Cho a,b,c0 thỏa mãn a^2+b^2+c^23
Chứng minh a, frac{1}{4-sqrt{ab}}+frac{1}{4-sqrt{bc}}+frac{1}{4-sqrt{ca}}le1
b, frac{2a^2}{a+b^2}+frac{2b^2}{b+c^2}+frac{2c^2}{c+a^2}ge a+b+c
3, Cho a,b,c 0 và frac{1}{a^2}+frac{1}{b^2}+frac{1}{c^2}1
Tính GTLN của P frac{1}{sqrt{5a^2+2ab+2b^2}}+frac{1}{sqrt{5b^2+2bc+2c^2}}+frac{1}{sqrt{...
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Ôn tập Bất đẳng thức
1 , Cho a,b,c<3 thỏa mãn abc(a+b+c)=3 . Tìm GTNN của C= \(\frac{a}{\sqrt{9-b^2}}+\frac{b}{\sqrt{9-c^2}}+\frac{c}{\sqrt{9-a^2}}\)
2, Cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=3\)
Chứng minh a, \(\frac{1}{4-\sqrt{ab}}+\frac{1}{4-\sqrt{bc}}+\frac{1}{4-\sqrt{ca}}\le1\)
b, \(\frac{2a^2}{a+b^2}+\frac{2b^2}{b+c^2}+\frac{2c^2}{c+a^2}\ge a+b+c\)
3, Cho a,b,c >0 và \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=1\)
Tính GTLN của P= \(\frac{1}{\sqrt{5a^2+2ab+2b^2}}+\frac{1}{\sqrt{5b^2+2bc+2c^2}}+\frac{1}{\sqrt{5c^2+2ca+2a^2}}\)
4 , Cho a,b,c>0 và \(ab+bc+ca\ge a+b+c\)
Chứng minh \(\frac{a^2}{\sqrt{a^3+8}}+\frac{b^2}{\sqrt{b^3+8}}+\frac{c^2}{\sqrt{c^3+8}}\ge1\)
3.
\(5a^2+2ab+2b^2=\left(a^2-2ab+b^2\right)+\left(4a^2+4ab+b^2\right)\)
\(=\left(a-b\right)^2+\left(2a+b\right)^2\ge\left(2a+b\right)^2\)
\(\Rightarrow\sqrt{5a^2+2ab+2b^2}\ge2a+b\)
\(\Rightarrow\frac{1}{\sqrt{5a^2+2ab+2b^2}}\le\frac{1}{2a+b}\)
Tương tự \(\frac{1}{\sqrt{5b^2+2bc+2c^2}}\le\frac{1}{2b+c};\frac{1}{\sqrt{5c^2+2ca+2a^2}}\le\frac{1}{2c+a}\)
\(\Rightarrow P\le\frac{1}{2a+b}+\frac{1}{2b+c}+\frac{1}{2c+a}\)
\(\le\frac{1}{9}\left(\frac{1}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}+\frac{1}{c}+\frac{1}{c}+\frac{1}{a}\right)\)
\(=\frac{1}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\le\frac{1}{3}.\sqrt{3\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)}=\frac{\sqrt{3}}{3}\)
\(\Rightarrow MaxP=\frac{\sqrt{3}}{3}\Leftrightarrow a=b=c=\sqrt{3}\)
1.\(\frac{a}{3}=\frac{b}{4}=\frac{c}{5}\) và \(^{a^2+2b^2-c^2}\)= 144
2.\(\frac{a}{5}=\frac{b}{7}=\frac{c}{9}\)và \(a^2+b^2-c^2\)=-28
1)
Ta có : \(\frac{a}{3}=\frac{b}{4}=\frac{c}{5}\)=> \(\frac{a^2}{9}=\frac{b^2}{16}=\frac{c^2}{25}\)=> \(\frac{a^2}{9}=\frac{2b^2}{32}=\frac{c^2}{25}\)
Đặt \(\frac{a^2}{9}=\frac{2b^2}{32}=\frac{c^2}{25}=k\)
=> \(\hept{\begin{cases}a^2=9k\\2b^2=32k\\c^2=25k\end{cases}}\)
=> \(a^2+2b^2-c^2=9k+32k-25k=16k\)
=> \(16k=144\)
=> \(k=9\)
Do đó \(\hept{\begin{cases}a^2=9\cdot9\\2b^2=32\cdot9\\c^2=25\cdot9\end{cases}}\Rightarrow\hept{\begin{cases}a^2=81\\b^2=144\\c^2=225\end{cases}}\Rightarrow\hept{\begin{cases}a=9\\b=12\\c=15\end{cases}}\)
2) Ta có : \(\frac{a}{5}=\frac{b}{7}=\frac{c}{9}\)=> \(\frac{a^2}{25}=\frac{b^2}{49}=\frac{c^2}{81}\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có :
\(\frac{a^2}{25}=\frac{b^2}{49}=\frac{c^2}{81}=\frac{a^2+b^2-c^2}{25+49-81}=\frac{-28}{-7}=4\)
=> \(\hept{\begin{cases}\frac{a^2}{25}=4\\\frac{b^2}{49}=4\\\frac{c^2}{81}=4\end{cases}}\Rightarrow\hept{\begin{cases}a^2=100\\b^2=196\\c^2=324\end{cases}}\Rightarrow\hept{\begin{cases}a=10\\b=14\\c=18\end{cases}}\)
a) đặt \(\frac{a}{3}=\frac{b}{4}=\frac{c}{5}=k\Rightarrow\hept{\begin{cases}a=3k\\b=4k\\c=5k\end{cases}}\)
đặt \(a^2+2b^2-c^2=144\)
\(\Leftrightarrow\left(3k\right)^2+2\left(4k\right)^2-\left(5k\right)^2=144\)
\(\Leftrightarrow9k^2+32k^2-25k^2=144\)
\(\Leftrightarrow k^2\left(9+32-25\right)=144\)
\(\Leftrightarrow k^216=144\)
\(\Leftrightarrow k^2=9\)
\(\Leftrightarrow k=\sqrt{9}=\pm3\)
do đó
\(\frac{a}{3}=k\Leftrightarrow\frac{a}{3}=\pm3\Rightarrow\hept{\begin{cases}a=3.3=9\\a=3.\left(-3\right)=-9\end{cases}}\)
\(\frac{b}{4}=k\Leftrightarrow\frac{b}{4}=\pm3\Rightarrow\hept{\begin{cases}b=4.3=12\\b=4.\left(-3\right)=-12\end{cases}}\)
\(\frac{c}{5}=k\Leftrightarrow\frac{c}{5}=\pm3\Rightarrow\hept{\begin{cases}c=5.3=15\\c=5.\left(-3\right)=-15\end{cases}}\)
vậy các cặp a,b,c thỏa mãn là \(\left\{a=9;b=12;c=15\right\}\left\{a=-9;b=-12;c=-15\right\}\)
bài 1,2 của anh sang sai nha bình phương ra phải có hai kết quả \(\pm\)chứ
2)đặt \(\frac{a}{5}=\frac{b}{7}=\frac{c}{9}=t\Rightarrow\hept{\begin{cases}a=5t\\b=7t\\c=9t\end{cases}}\)
ta có \(a^2+b^2-c^2=-28\)
\(\Leftrightarrow\left(5t\right)^2+\left(7t\right)^2-\left(9t\right)^2=-28\)
\(\Leftrightarrow25t^2+49t^2-81t^2=-28\)
\(\Leftrightarrow t^2\left(25+49-81\right)=-28\)
\(\Leftrightarrow t^2\left(-7\right)=-28\)
\(\Leftrightarrow t^2=4\)
\(\Leftrightarrow t=\sqrt{4}=\pm2\)
do đó
\(\frac{a}{5}=t\Leftrightarrow\frac{a}{5}=\pm2\Rightarrow\hept{\begin{cases}a=5.2=10\\a=5.\left(-2\right)=-10\end{cases}}\)
\(\frac{b}{7}=t\Leftrightarrow\frac{b}{7}=\pm2\Rightarrow\hept{\begin{cases}b=7.2=14\\b=7.\left(-2\right)=-14\end{cases}}\)
\(\frac{c}{9}=t\Leftrightarrow\frac{c}{9}=\pm2\Rightarrow\hept{\begin{cases}c=9.2=18\\c=9.\left(-2\right)=-18\end{cases}}\)
vậy các cặp a,b,c thỏa mãn là \(\left\{a=10;b=14;=18\right\}\left\{a=-10;b=-14;b=-18\right\}\)
áp dụng cô si ta có:+)frac{a^5}{b^3}+frac{a^3}{b}gefrac{2a^4}{b^2};frac{b^5}{c^3}+frac{b^3}{c}gefrac{2b^4}{c^2};frac{c^5}{a^3}+frac{c^3}{a}gefrac{2c^4}{a^2}Leftrightarrowfrac{a^5}{b^3}+frac{b^5}{c^3}+frac{c^5}{a^3}ge2left(frac{a^4}{b^2}+frac{b^4}{c^2}+frac{c^4}{a^2}right)-left(frac{a^3}{b}+frac{b^3}{c}+frac{c^3}{a}right)+)frac{a^4}{b^2}+a^2gefrac{2a^3}{b};frac{b^4}{c^2}+b^2gefrac{2b^3}{c};frac{c^4}{a^2}+c^2gefrac{2C^3}{a}Leftrightarrowfrac{a^4}{b^2}+frac{b^4}{c^2}+frac{c^4}{a^2}ge2left(frac{a^3}...
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áp dụng cô si ta có:
+)\(\frac{a^5}{b^3}+\frac{a^3}{b}\ge\frac{2a^4}{b^2};\frac{b^5}{c^3}+\frac{b^3}{c}\ge\frac{2b^4}{c^2};\frac{c^5}{a^3}+\frac{c^3}{a}\ge\frac{2c^4}{a^2}\)
\(\Leftrightarrow\frac{a^5}{b^3}+\frac{b^5}{c^3}+\frac{c^5}{a^3}\ge2\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)-\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)\)
+)\(\frac{a^4}{b^2}+a^2\ge\frac{2a^3}{b};\frac{b^4}{c^2}+b^2\ge\frac{2b^3}{c};\frac{c^4}{a^2}+c^2\ge\frac{2C^3}{a}\)
\(\Leftrightarrow\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\ge2\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)-\left(a^2+b^2+c^2\right)\)
+)\(\frac{a^3}{b}+ab\ge2a^2;\frac{b^3}{c}+bc\ge2b^2;\frac{c^3}{a}+ca\ge2c^2\)
\(\Leftrightarrow\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\ge\left(a^2+b^2+c^2\right)+\left(a^2+b^2+c^2-ab-bc-ca\right)\ge\left(a^2+b^2+c^2\right)\)
\(\Leftrightarrow\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\ge\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)+\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-a^2-b^2-c^2\right)\ge\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\)
\(\Leftrightarrow\frac{a^5}{b^3}+\frac{b^5}{c^3}+\frac{c^5}{a^3}\ge\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)+\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}-\frac{a^3}{b}-\frac{b^3}{c}-\frac{c^3}{a}\right)\ge\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)\)
Cho a, b, c > 0 và a + b + c = 3. Tìm GTNN của:
a, \(A=\frac{a^2}{a+1}+\frac{b^2}{b+1}+\frac{c^2}{c+1}\)
b, \(B=\frac{1}{a}+\frac{4}{b+1}+\frac{9}{c+2}\)
c, \(C=\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\)
d, \(D=a^4+b^4+c^4+a^2+b^2+c^2+2020\)
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Câu hỏi của Nobody - Toán lớp 8 | Học trực tuyến
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Cho a,b,c 0.Chứng minh rằnga,frac{1}{a}+frac{1}{b}+frac{1}{c}gefrac{2}{a+b}+frac{2}{b+c}+frac{2}{c+a}b,frac{4}{a}+frac{5}{b}+frac{3}{c}ge4left(frac{3}{a+b}+frac{2}{b+c}+frac{1}{c+a}right)
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Cho a,b,c > 0.Chứng minh rằng
a,\(\frac{1}{a}\)+\(\frac{1}{b}\)+\(\frac{1}{c}\)\(\ge\)\(\frac{2}{a+b}\)+\(\frac{2}{b+c}\)+\(\frac{2}{c+a}\)
b,\(\frac{4}{a}\)+\(\frac{5}{b}\)+\(\frac{3}{c}\)\(\ge\)\(4\left(\frac{3}{a+b}+\frac{2}{b+c}+\frac{1}{c+a}\right)\)
Ta chứng minh BĐT sau với các số dương:
\(\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{4}{x+y}\)
Thật vậy, BĐT tương đương: \(\dfrac{x+y}{xy}\ge\dfrac{4}{x+y}\Leftrightarrow\left(x+y\right)^2\ge4xy\)
\(\Leftrightarrow x^2-2xy+y^2\ge0\Leftrightarrow\left(x-y\right)^2\ge0\) (luôn đúng)
Áp dụng:
\(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\) ; \(\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{4}{b+c}\) ; \(\dfrac{1}{c}+\dfrac{1}{a}\ge\dfrac{4}{c+a}\)
Cộng vế với vế:
\(2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\ge\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a}\)
\(\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{2}{a+b}+\dfrac{2}{b+c}+\dfrac{2}{c+a}\)
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b.
Ta có:
\(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\Rightarrow\dfrac{3}{a}+\dfrac{3}{b}\ge\dfrac{12}{a+b}\) (1)
\(\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{4}{b+c}\Rightarrow\dfrac{2}{b}+\dfrac{2}{c}\ge\dfrac{8}{b+c}\) (2)
\(\dfrac{1}{c}+\dfrac{1}{a}\ge\dfrac{4}{c+a}\) (3)
Cộng vế với vế (1); (2) và (3):
\(\dfrac{4}{a}+\dfrac{5}{b}+\dfrac{3}{c}\ge4\left(\dfrac{3}{a+b}+\dfrac{2}{b+c}+\dfrac{1}{c+a}\right)\) (đpcm)
Dấu "=" xảy ra khi \(a=b=c\)
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Cho a,b,c 0.Chứng minh rằnga,frac{1}{a}+frac{1}{b}+frac{1}{c}gefrac{2}{a+b}+frac{2}{b+c}+frac{2}{c+a}b,frac{4}{a}+frac{5}{b}+frac{3}{c}ge4left(frac{3}{a+b}+frac{2}{b+c}+frac{1}{c+a}right)
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Cho a,b,c > 0.Chứng minh rằng
a,\(\frac{1}{a}\)+\(\frac{1}{b}\)+\(\frac{1}{c}\)\(\ge\)\(\frac{2}{a+b}\)+\(\frac{2}{b+c}\)+\(\frac{2}{c+a}\)
b,\(\frac{4}{a}\)+\(\frac{5}{b}\)+\(\frac{3}{c}\)\(\ge\)\(4\left(\frac{3}{a+b}+\frac{2}{b+c}+\frac{1}{c+a}\right)\)
biết \(\frac{a}{2}=\frac{b}{3};\frac{a}{4}=\frac{c}{9}\)và \(a^3+b^3+c^3=-1009\).tính a,b,c
Ta có :
\(\frac{a}{2}=\frac{b}{3};\frac{a}{4}=\frac{c}{9}\)
\(\Rightarrow\frac{a}{4}=\frac{b}{6}=\frac{c}{9}\)
\(\Rightarrow\frac{a^3}{64}=\frac{b^3}{216}=\frac{c^3}{729}\)
Áp dụng c/t tỉ lệ thức = nhau ta có :
\(\frac{a^3}{64}=\frac{b^3}{216}=\frac{c^3}{729}=\frac{a^3+b^3+c^3}{64+216+729}=\frac{-1009}{1009}=-1\)
\(\frac{a^3}{64}=-1\Rightarrow a^3=-64\Rightarrow a=-4\)\(\frac{b^3}{216}=-1\Rightarrow b^3=-216\Rightarrow a=-6\)\(\frac{c^3}{729}=-1\Rightarrow c^3=-729\Rightarrow a=-9\)Vậy a = -4 b = -6 c = -9
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