tìm giá trị lớn nhất của biểu thức:
a) A = 5 + \(\dfrac{15}{4\left|3x+7\right|+3}\)
b) B = \(\dfrac{-1}{3}+\dfrac{21}{8\left|15x-21\right|+7}\)
hãy tính giá trị của biểu thức sau:
C=\(\left|-3\left(-\dfrac{13}{15}-\dfrac{17}{21}\right)\right|-\left|-\dfrac{13}{5}+\dfrac{17}{7}\right|+\left(-12+\dfrac{35}{3}\right):\left|-\dfrac{7}{6}\right|\)
\(C=\left|-3\left(\dfrac{-13}{15}-\dfrac{17}{21}\right)\right|-\left|\dfrac{-13}{15}+\dfrac{17}{7}\right|+\left(-12+\dfrac{35}{3}\right):\left|-\dfrac{7}{6}\right|\\ =\left|-3.-\dfrac{176}{105}\right|-\left|-\dfrac{6}{35}\right|+\left(-\dfrac{1}{3}\right):\dfrac{7}{6}\\ =\dfrac{176}{35}-\dfrac{6}{35}-\dfrac{1}{3}:\dfrac{7}{6}\\ =\dfrac{176}{35}-\dfrac{6}{35}-\dfrac{2}{7}\\ =\dfrac{170}{35}-\dfrac{2}{7}=\dfrac{32}{7}.\)
Tìm giá trị lớn nhất , giá trị nhỏ nhất của biểu thức :
a)\(A=\left|\dfrac{3}{5}-x\right|+\dfrac{1}{9}\)
b)B=\(\dfrac{2009}{2008}-\left|x-\dfrac{3}{5}\right|\)
c)C=\(-2\left|\dfrac{1}{3}x+4\right|+1\dfrac{2}{3}\)
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\(A=\left|\dfrac{3}{5}-x\right|+\dfrac{1}{9}\ge\dfrac{1}{9}\\ A_{min}=\dfrac{1}{9}\Leftrightarrow x=\dfrac{3}{5}\\ B=\dfrac{2009}{2008}-\left|x-\dfrac{3}{5}\right|\le\dfrac{2009}{2008}\\ B_{max}=\dfrac{2009}{2008}\Leftrightarrow x=\dfrac{3}{5}\\ C=-2\left|\dfrac{1}{3}x+4\right|+1\dfrac{2}{3}\le1\dfrac{2}{3}\\ C_{max}=1\dfrac{2}{3}\Leftrightarrow\dfrac{1}{3}x=-4\Leftrightarrow x=-12\)
a: \(A=\left|\dfrac{3}{5}-x\right|+\dfrac{1}{9}\ge\dfrac{1}{9}\forall x\)
Dấu '=' xảy ra khi \(x=\dfrac{3}{5}\)
Tính giá trị của các biểu thức sau :
a)\(\left(7+3\dfrac{1}{4}-\dfrac{3}{5}\right)\)+(0,4 - 5) - \(\left(4\dfrac{1}{4}-1\right)\)
b)\(\dfrac{2}{3}\) - \(\left[\left(-\dfrac{7}{4}\right)-\left(\dfrac{1}{2}+\dfrac{3}{8}\right)\right]\)
c)\(\left(9-\dfrac{1}{2}-\dfrac{3}{4}\right)\):\(\left(7-\dfrac{1}{4}-\dfrac{5}{8}\right)\)
d)3 - \(\dfrac{1-\dfrac{1}{7}}{1+\dfrac{1}{7}}\)
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\(a,\left(7+3\dfrac{1}{4}-\dfrac{3}{5}\right)+\left(0,4-5\right)-\left(4\dfrac{1}{4}-1\right)\)
\(=\left(7+\dfrac{13}{4}-\dfrac{3}{5}\right)-\dfrac{23}{5}-\left(\dfrac{17}{4}-1\right)\)
\(=7+\dfrac{13}{4}-\dfrac{3}{5}-\dfrac{23}{5}-\dfrac{17}{4}+1\)
\(=\left(7+1\right)+\left(\dfrac{13}{4}-\dfrac{17}{4}\right)-\left(\dfrac{3}{5}+\dfrac{23}{5}\right)\)
\(=8-\dfrac{4}{4}-\dfrac{26}{5}\)
\(=7-\dfrac{26}{5}\)
\(=\dfrac{9}{5}\)
\(b,\dfrac{2}{3}-\left[\left(-\dfrac{7}{4}\right)-\left(\dfrac{1}{2}+\dfrac{3}{8}\right)\right]\)
\(=\dfrac{2}{3}-\left(-\dfrac{7}{4}-\dfrac{1}{2}-\dfrac{3}{8}\right)\)
\(=\dfrac{2}{3}-\left(-\dfrac{14}{8}-\dfrac{4}{8}-\dfrac{3}{8}\right)\)
\(=\dfrac{2}{3}-\left(-\dfrac{21}{8}\right)\)
\(=\dfrac{2}{3}+\dfrac{21}{8}\)
\(=\dfrac{79}{24}\)
\(c,\left(9-\dfrac{1}{2}-\dfrac{3}{4}\right):\left(7-\dfrac{1}{4}-\dfrac{5}{8}\right)\)
\(=\left(\dfrac{36}{4}-\dfrac{2}{4}-\dfrac{3}{4}\right):\left(\dfrac{56}{8}-\dfrac{2}{8}-\dfrac{5}{8}\right)\)
\(=\dfrac{31}{4}:\dfrac{49}{8}\)
\(=\dfrac{62}{49}\)
\(d,3-\dfrac{1-\dfrac{1}{7}}{1+\dfrac{1}{7}}=3-\dfrac{\dfrac{7}{7}-\dfrac{1}{7}}{\dfrac{7}{7}+\dfrac{1}{7}}=3-\left(\dfrac{6}{7}:\dfrac{8}{7}\right)=3-\dfrac{3}{4}=\dfrac{9}{4}\)
Tính giá trị của biểu thức:
a) (-7,05 - \(\dfrac{1}{4}\)) : (-5) + \(\dfrac{1}{15}\) - \(\left(-\dfrac{1}{5}\right)\) : (-3)
b) \(\left(\dfrac{3}{25}-1,12\right)\) : \(\dfrac{3}{7}\) : [\(\left(3\dfrac{1}{2}\right)\) - \(\left(3\dfrac{2}{3}\right)\) : \(\dfrac{1}{14}\)]
a) = (\(-\dfrac{141}{20}\)- \(\dfrac{1}{4}\)) : (-5) + \(\dfrac{1}{15}\) - \(\dfrac{1}{15}\)
= \(-\dfrac{73}{10}\) : - 5
= \(\dfrac{73}{50}\)
b) = \(\left(\dfrac{3}{25}-\dfrac{28}{25}\right)\). \(\dfrac{7}{3}\) : \(\left(\dfrac{7}{2}-\dfrac{11}{3}.14\right)\)
= \(-\dfrac{7}{3}\) . \(-\dfrac{6}{287}\)
= \(\dfrac{2}{41}\)
Tính giá trị các biểu thức sau theo cách hợp lí nhất.
a) $A=\left(\dfrac{1}{3}-\dfrac{8}{15}-\dfrac{1}{7}\right)+\left(\dfrac{2}{3}+\dfrac{-7}{15}+1 \dfrac{1}{7}\right)$;
b) $\mathrm{B}=0,25+\dfrac{3}{5}-\left(\dfrac{1}{8}-\dfrac{2}{5}+1 \dfrac{1}{4}\right)$.
a) \(A=\left(\dfrac{1}{3}-\dfrac{8}{15}-\dfrac{1}{7}\right)+\left(\dfrac{2}{3}+\dfrac{-7}{15}+1\dfrac{1}{7}\right)\)
\(A=\dfrac{1}{3}-\dfrac{8}{15}-\dfrac{1}{7}+\dfrac{2}{3}+\dfrac{-7}{15}+1+\dfrac{1}{7}\)
\(A=\left(\dfrac{1}{3}+\dfrac{2}{3}\right)+\left(\dfrac{-8}{15}+\dfrac{-7}{15}\right)+\left(\dfrac{-1}{7}+\dfrac{1}{7}\right)+1\)
\(A=1+\left(-1\right)+0+1=\left(1-1\right)+1\)
\(A=1\)
b) \(B=0,25+\dfrac{3}{5}-\left(\dfrac{1}{8}-\dfrac{2}{5}+1\dfrac{1}{4}\right)\)
\(B=\dfrac{1}{4}+\dfrac{3}{5}-\dfrac{1}{8}+\dfrac{2}{5}-1-\dfrac{1}{4}\)
\(B=\left(\dfrac{1}{4}-\dfrac{1}{4}\right)+\left(\dfrac{3}{5}+\dfrac{2}{5}\right)-1-\dfrac{1}{8}\)
\(B=0+1-1-\dfrac{1}{8}\)
\(B=-\dfrac{1}{8}\)
tính giá trị biểu thức sau
a) \(A=\dfrac{9^4}{3^2}\)
b) \(B=81.\left(\dfrac{5}{3}\right)^4\)
c) \(C=\left(\dfrac{4}{7}\right)^{-4}.\left(\dfrac{2}{7}\right)^3\)
d) \(D=7^{-6}.\left(\dfrac{2}{3}\right)^0.\left(\dfrac{7}{5}\right)^6\)
e) \(E=8^3:\left(\dfrac{2}{3}\right)^5.\left(\dfrac{1}{3}\right)^2\)
f) \(F=\left(\dfrac{7}{9}\right)^{-2}.\left(\dfrac{1}{\sqrt{3}}\right)^8\)
g) \(G=\left(\dfrac{-4}{5}\right)^{-2}.\left(\dfrac{2}{5}\right)^2.\left(\sqrt{2}\right)^3\)
a: \(A=\dfrac{9^4}{3^2}=\dfrac{\left(3^2\right)^4}{3^2}=\dfrac{3^8}{3^2}=3^6\)=729
b: \(B=81\left(\dfrac{5}{3}\right)^4=81\cdot\dfrac{5^4}{3^4}=\dfrac{81}{3^4}\cdot5^4=5^4=625\)
c: \(C=\left(\dfrac{4}{7}\right)^{-4}\cdot\left(\dfrac{2}{7}\right)^3\)
\(=\left(\dfrac{7}{4}\right)^4\cdot\left(\dfrac{2}{7}\right)^3\)
\(=\dfrac{7^4}{4^4}\cdot\dfrac{2^3}{7^3}\)
\(=\dfrac{2^3}{4^4}\cdot7\)
\(=\dfrac{2^3}{2^8}\cdot7=\dfrac{7}{2^5}=\dfrac{7}{32}\)
d: \(D=7^{-6}\cdot\left(\dfrac{2}{3}\right)^0\left(\dfrac{7}{5}\right)^6\)
\(=7^{-6}\left(\dfrac{7}{5}\right)^6\)
\(=\dfrac{1}{7^6}\cdot\dfrac{7^6}{5^6}=\dfrac{1}{5^6}=\dfrac{1}{15625}\)
e: \(E=8^3:\left(\dfrac{2}{3}\right)^5\cdot\left(\dfrac{1}{3}\right)^2\)
\(=2^6:\dfrac{2^5}{3^5}\cdot\dfrac{1}{3^2}\)
\(=2^6\cdot\dfrac{3^5}{2^5}\cdot\dfrac{1}{3^2}\)
\(=\dfrac{2^6}{2^5}\cdot\dfrac{3^5}{3^2}=3^3\cdot2=54\)
f: \(F=\left(\dfrac{7}{9}\right)^{-2}\cdot\left(\dfrac{1}{\sqrt{3}}\right)^8\)
\(=\left(\dfrac{9}{7}\right)^2\cdot\left(\dfrac{1}{3}\right)^4\)
\(=\dfrac{9^2}{7^2}\cdot\dfrac{1}{3^4}=\dfrac{9^2}{3^4}\cdot\dfrac{1}{7^2}=\dfrac{81}{81}\cdot\dfrac{1}{49}=\dfrac{1}{49}\)
g: \(G=\left(-\dfrac{4}{5}\right)^{-2}\cdot\left(\dfrac{2}{5}\right)^2\cdot\left(\sqrt{2}\right)^3\)
\(=\left(-\dfrac{5}{4}\right)^2\cdot\left(\dfrac{2}{5}\right)^2\cdot2\sqrt{2}\)
\(=\dfrac{25}{16}\cdot\dfrac{4}{25}\cdot2\sqrt{2}=\dfrac{4}{16}\cdot2\sqrt{2}=\dfrac{8\sqrt{2}}{16}=\dfrac{\sqrt{2}}{2}\)
Tính giá trị của biểu thức:
\(A=\dfrac{-3}{7}.\dfrac{5}{9}+\dfrac{4}{9}.\dfrac{-3}{7}+\left(-2022\right)^0\)
\(B=0,75-\left(2\dfrac{1}{3}+0,75\right)+3^2.\left(-\dfrac{1}{9}\right)\)
\(C=2\dfrac{6}{7}.\left[\left(\dfrac{-7}{5}-\dfrac{3}{2}:\dfrac{-5}{-4}\right)+\left(\dfrac{3}{2}\right)^2\right]\)
\(D=\dfrac{2}{7}+\dfrac{5}{7}.\left(\dfrac{3}{5}-0,25\right).\left(-2\right)^2+35\%\)
\(E=1\dfrac{13}{15}.0,75-\left(\dfrac{11}{20}+25\%\right):1\dfrac{2}{5}\)
\(F=\dfrac{\dfrac{5}{3}-\dfrac{5}{7}+\dfrac{5}{9}}{\dfrac{10}{3}-\dfrac{10}{7}+\dfrac{10}{9}}\)
1. Cho số nguyên dương x, tìm giá trị nhỏ nhất của biểu thức:
\(P=\dfrac{\left(x+1\right)^6}{\left(x^3+7\right)\left(x^3+3x^2+4\right)}\).
2. Cho \(a,b\ge0\) thỏa mãn \(a-\sqrt{a}=\sqrt{b}-b\), tìm giá trị nhỏ nhất của biểu thức:
\(M=\left(a-b\right)\left(a+b-1\right)\).
3. Cho \(\Delta OEF\) vuông tại O có \(OE=a\), \(OF=b\), \(EF=c\) và \(\widehat{OEF}=\alpha\), \(\widehat{OFE}=\beta\).
1)
i, Chứng minh rằng không có giá trị nào của a,b,c để biểu thức \(A=\dfrac{a+b}{c}+\dfrac{c}{a+b}\) nhận giá trị nguyên.
ii, Giả sử \(c\sqrt{ab}=\sqrt{2}\) , tìm giá trị nhỏ nhất của biểu thức \(B=\left(a+b\right)^2\).
2)
i, Tìm giá trị nhỏ nhất của biểu thức \(C=\dfrac{1}{\sin^2\alpha}+\dfrac{1}{\sin^2\beta}-2\left(\sin^2\alpha+\sin^2\beta\right)+\dfrac{\sin\alpha}{\tan\alpha}-\dfrac{\tan\alpha+\cos\beta}{\cot\beta}\) .
ii, Tìm điều kiện của \(\Delta OEF\) khi \(2\cos^2\beta-\cot^2\alpha+\dfrac{1}{\sin^2\alpha}=2\).
Tìm giá trị lớn nhất của biểu thức :
\(B=\frac{-1}{3}+\frac{21}{8.\left|15x-21\right|+7}\)
B đạt giá trị lớn nhất \(\Leftrightarrow\frac{21}{8.\left|15x-21\right|+7}\) đạt GTLN
\(\Leftrightarrow8.\left|15x-21\right|+7\) đạt GTNN
Vì \(\left|15x-21\right|\ge0\left(\forall x\in Z\right)\)
Nên suy ra \(8.\left|15x-21\right|+7\ge7\)
Dấu "=" xảy ra <=> \(15x-21=0\Leftrightarrow15x=21\Leftrightarrow x=\frac{21}{15}=\frac{7}{5}\)
Vậy GTLN của biểu thức B = \(\frac{-1}{3}+\frac{21}{7}=\frac{8}{3}\) khi \(x=\frac{7}{5}\)
\(B=-\frac{1}{3}+\frac{21}{8\left|15x-21\right|+7}\le-\frac{1}{3}+\frac{21}{7}=-\frac{1}{3}+3=\frac{8}{3}\)
Dấu ''='' xảy ra \(\Leftrightarrow15x-21=0\)
\(\Leftrightarrow x=\frac{7}{5}\)
Vậy ........