Tính \(S=\sqrt[3]{26+15\sqrt{3}}\left(2-\sqrt{3}\right)+\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\)
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Cho \(x=\dfrac{\sqrt[3]{26+15\sqrt{3}}.\left(2-\sqrt{3}\right)}{\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}}\). Tính giá trị của biểu thức: \(M=\left(3x^3-x^2-1\right)^{2021}\)
\(x=\dfrac{\sqrt[3]{\left(2+\sqrt{3}\right)^3}\left(2-\sqrt{3}\right)}{\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}}=\dfrac{1}{\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}}\)
Đặt \(A=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)\(\Leftrightarrow A^3=18+3\sqrt[3]{\left(9-4\sqrt{5}\right)\left(9+4\sqrt{5}\right)}\left(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\right)\\ \Leftrightarrow A^3=18+3A\sqrt[3]{1}\\ \Leftrightarrow A^3-3A-18=0\\ \Leftrightarrow A=3\\ \Leftrightarrow X=\dfrac{1}{3}\\ \Leftrightarrow Q=\left[3\left(\dfrac{1}{3}\right)^3-\left(\dfrac{1}{3}\right)^2-1\right]^{2021}=\left(\dfrac{1}{9}-\dfrac{1}{9}-1\right)^{2021}=\left(-1\right)^{2021}=-1\)
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Tính:
\(\dfrac{\sqrt[3]{26+15\sqrt{3}}\left(2-\sqrt{3}\right)}{\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}}\)
Đặt A = \(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\)=> \(A^3=18+3A\Leftrightarrow A^3-3A-18=0\Leftrightarrow\left(A-3\right)\left(A^2+3A+6\right)=0\Leftrightarrow A-3=0\Leftrightarrow A=3\)
\(\dfrac{\sqrt[3]{26+15\sqrt{3}}\left(2-\sqrt{3}\right)}{\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}}=\dfrac{\sqrt[3]{\left(2+\sqrt{3}\right)^3}\left(2-\sqrt{3}\right)}{3}=\dfrac{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}{3}=\dfrac{1}{3}\)
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Rút gọn biểu thức:
\(M=\sqrt[3]{26+15\sqrt{3}}.\left(2-\sqrt{3}\right)+\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\)
Rút gọn biểu thức
\(M=\sqrt[3]{26+15\sqrt{3}}.\left(2-\sqrt{3}\right)+\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\)
Ta có \(\sqrt[3]{26+15\sqrt{3}}=\sqrt[3]{8+12\sqrt{3}+18+3\sqrt{3}}\)
\(=\sqrt[3]{2^3+3.2^2\sqrt{3}+3.2.\left(\sqrt{3}\right)^2+\left(\sqrt{3}\right)^3}=\sqrt[3]{\left(2+\sqrt{3}\right)^3}\)
\(=2+\sqrt{3}\)
Đặt \(x=\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\)
Ta có \(x^3=\left(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\right)^3\)
\(=9+\sqrt{80}+9-\sqrt{80}+3.\left(\sqrt[3]{9+\sqrt{80}}\right)^2\left(\sqrt[3]{9-\sqrt{80}}\right)+3.\left(\sqrt[3]{9-\sqrt{80}}\right)^2\left(\sqrt[3]{9+\sqrt{80}}\right)\)
\(=18+3\sqrt[3]{9+\sqrt{80}}.\sqrt[3]{9-\sqrt{80}}\left(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\right)\)
\(=18+3\sqrt[3]{9^2-80}.x\)
\(=18+3x\)
Vậy \(x^3=18+3x\)
\(\Leftrightarrow x^3-3x-18=0\)
Vậy x = 3
Do đó \(M=\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)+3=2^2-3+3=4\)
Vậy M = 4.
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Cho \(x=\frac{\sqrt[3]{26+15\sqrt{3}}\left(2-\sqrt{3}\right)}{\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}}\)
tính \(R=\left(3x^3-x^2-1\right)^{2004}\)
cho \(x=\frac{26+15\sqrt{3}}{\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}}}\)
tính \(\left(3x^3+x^2+1\right)^{2018}\)
Tính
a.A=\(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}\)
b.B=\(\sqrt[3]{3+\sqrt{9+\dfrac{125}{7}}}-\sqrt[3]{-3+\sqrt{9+\dfrac{125}{7}}}\)
c.C=\(\sqrt[3]{26+15\sqrt{3}}.\left(2-\sqrt{3}\right)+\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}\)
a: \(A^3=2+\sqrt{5}+2-\sqrt{5}+3\cdot A\cdot\sqrt[3]{4-5}\)
\(\Leftrightarrow A^3=4-3A\)
=>A=1
c: \(C=1+\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)
\(=1+3=4\)
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Tính GTBT
a,M=\(\left(3x^3-x^2-1\right)^{2018}\) biết x = \(\dfrac{\sqrt[3]{26+15\sqrt{3}}\left(2-\sqrt[]{3}\right)}{\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}}\)
b,\(x^3+ax+b\) biết x=\(\sqrt[3]{\dfrac{-b}{2}+\sqrt{\dfrac{b^2}{4}+\dfrac{a^3}{27}}}+\sqrt[3]{\dfrac{-b}{2}-\sqrt{\dfrac{b^2}{4}+\dfrac{a^3}{27}}}\)
Tính
1, a = \(\sqrt[3]{45+26\sqrt{2}}+\sqrt[3]{45-29\sqrt{2}}\)
2, x = \(\sqrt[3]{4+\sqrt{80}-\sqrt[3]{\sqrt{80}-4}}\)
3, \(\left(4+\sqrt{15}\right)\cdot\left(\sqrt{10}-\sqrt{6}\right)\cdot\sqrt{4-\sqrt{15}}\)
4, \(\sqrt{4-\sqrt{7}}-\sqrt{4+\sqrt{7}}\)
5, \(\sqrt{\frac{4-\sqrt{7}}{4+\sqrt{7}}}+\sqrt{\frac{4+\sqrt{7}}{4-\sqrt{7}}}\)
