a) chứng tỏ \(M=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\)là một số nguyên
b) tính \(A=\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\)
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1) \(A=2\sqrt{5}-6\sqrt{2}+3\sqrt{5}=5\sqrt{5}-6\sqrt{2}\)
2) \(B=\dfrac{30\left(\sqrt{7}+1\right)}{7-1}+\dfrac{15\left(\sqrt{7}-2\right)}{7-4}=5\sqrt{7}+5+5\sqrt{7}-10=-5+10\sqrt{7}\)
3) \(C=\left(3-\dfrac{\sqrt{5}\left(\sqrt{5}-1\right)}{\sqrt{5}-1}\right)\left(3+\dfrac{\sqrt{5}\left(\sqrt{5}+1\right)}{\sqrt{5}+1}\right)=\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)=9-5=4\)
4) \(D=3-\sqrt{2}+1-\sqrt{2}=4-2\sqrt{2}\)
Ta có:
\(R=\)\(\dfrac{3+\sqrt{5}}{2\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{3-\sqrt{5}}{2\sqrt{2}-\sqrt{3-\sqrt{5}}}\)
\(=\)\(\dfrac{\sqrt{10}+3\sqrt{2}}{5+\sqrt{5}}+\dfrac{\sqrt{10}-3\sqrt{2}}{5-\sqrt{5}}\)
\(=\dfrac{4\sqrt{2}}{\sqrt{5}\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}\)
\(=\dfrac{4\sqrt{2}}{4\sqrt{5}}=\sqrt{\dfrac{2}{5}}\)
Làm câu S tương tự như này rồi đối chiếu kết quả nha
\(1,=20-7=13\\ b,=12-50=-38\\ c,=\sqrt{7}-2+\sqrt{7}+2=2\sqrt{7}\\ d,=\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}=2\sqrt{3}\\ e,=11+2\sqrt{30}\\ f,=8-2\sqrt{15}\\ g,=11+2\sqrt{6}\)
1) \(=\left(2\sqrt{5}\right)^2-\left(\sqrt{7}\right)^2=20-7=13\)
2) \(=\left(2\sqrt{3}\right)^2-\left(5\sqrt{2}\right)^2=12-50=-38\)
3) \(=\sqrt{7}-2+\sqrt{7}+2=2\sqrt[]{7}\)
4) \(=\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}=2\sqrt{3}\)
5) \(=5+6-2\sqrt{5.6}=11-2\sqrt{30}\)
6) \(=3+5-2\sqrt{3.5}=8-4\sqrt{2}\)
7) \(=\left(2\sqrt{2}\right)^2+\left(\sqrt{3}\right)^2+2\sqrt{2\sqrt{2}.3}=11+2\sqrt{6\sqrt{2}}\)
\(A^3=\left(\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}-7}\right)^3\)
\(=\left(5\sqrt{2}+7\right)-\left(5\sqrt{2}-7\right)-3\sqrt[3]{5\sqrt{2}+7}.\sqrt[3]{5\sqrt{2}-7}\left(\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}+7}\right)\)
\(=14-3A\)
=> \(A^3+3A-14=0\)
<=> \(\left(A^3-8\right)+\left(3A-6\right)=0\)
<=> \(\left(A-2\right)\left(A^2+2A+7\right)=0\)
<=> A = 2 vì A^2 + 2A + 7 = (A+ 1) ^2 + 6 > 0
Do đó A là 1 số nguyên.
\(M^3=\sqrt{5}+2-\left(\sqrt{5}-2\right)-3\sqrt[3]{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}\left(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\right)\)
\(\Rightarrow M^3=4-3M\)
\(\Leftrightarrow\left(M-1\right)\left(M^2+M+4\right)\)
\(\Leftrightarrow M=1\text{ (do }M^2+M+4=\left(M+\frac{1}{2}\right)^2+\frac{15}{4}>0\text{)}\)
Vậy M = 1
b/ làm tương tự