Tính:
D=\(\left(\sqrt{57}+3\sqrt{6}+\sqrt{38}+6\right)\left(\sqrt{57}-3\sqrt{6}-\sqrt{38}+6\right)\)
PQ
Những câu hỏi liên quan
chứng minh rằng các biểu thức sau có giá trị là số nguyên
a. A=\(\left(\sqrt{57}+3\sqrt{6}+\sqrt{38}+6\right)\left(\sqrt{57}-3\sqrt{6}-\sqrt{38}+6\right)\)
b B=\(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)
c. C=\(\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)
- Cô giáo giải hộ em vs ạ
- Em cảm ơn
a) \(A=\left(\sqrt{57}+3\sqrt{6}+\sqrt{38}+6\right)\left(\sqrt{57}-3\sqrt{6}-\sqrt{38}+6\right)\)\(\Leftrightarrow A=\left[\left(\sqrt{57}+6\right)+\left(3\sqrt{6}+\sqrt{38}\right)\right]\left[\left(\sqrt{57}+6\right)-\left(3\sqrt{6}+\sqrt{38}\right)\right]\)\(\Leftrightarrow A=\left(\sqrt{57}+6\right)^2-\left(3\sqrt{6}+\sqrt{38}\right)^2\)
\(\Leftrightarrow A=57+12\sqrt{57}+36-54-12\sqrt{57}-38\)
\(\Leftrightarrow A=1\)
b) \(B=\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)\(\Leftrightarrow B=\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+4\sqrt{3}}}}}{\sqrt{6}+\sqrt{2}}\)\(\Leftrightarrow B=\dfrac{2\sqrt{3+\sqrt{5-\sqrt{1+4\sqrt{3}+\left(2\sqrt{3}\right)^2}}}}{\sqrt{6}+\sqrt{2}}\)\(\Leftrightarrow B=\dfrac{2\sqrt{3+\sqrt{5-\sqrt{\left(1+2\sqrt{3}\right)^2}}}}{\sqrt{6}+\sqrt{2}}\)
\(\Leftrightarrow B=\dfrac{2\sqrt{3+\sqrt{4-2\sqrt{3}}}}{\sqrt{6}+\sqrt{2}}\)
\(\Leftrightarrow B=\dfrac{2\sqrt{3+\sqrt{\left(\sqrt{3}-1\right)^2}}}{\sqrt{6}+\sqrt{2}}\)
\(\Leftrightarrow B=\dfrac{2\sqrt{2+\sqrt{3}}}{\sqrt{6}+\sqrt{2}}\)
\(\Leftrightarrow B=\dfrac{\sqrt{8+4\sqrt{3}}}{\sqrt{6}+\sqrt{2}}\)
\(\Leftrightarrow B=\dfrac{\sqrt{\left(\sqrt{6}+\sqrt{2}\right)^2}}{\sqrt{6}+\sqrt{2}}\)
\(\Leftrightarrow B=\dfrac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}}=1\)
c)\(C=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)
\(\Leftrightarrow C=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{3^2-2\times3\times2\sqrt{5}+\left(2\sqrt{5}\right)^2}}}\)
\(\Leftrightarrow C=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{\left(2\sqrt{5}-3\right)^2}}}\)
\(\Leftrightarrow C=\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}\)
\(\Leftrightarrow C=\sqrt{\sqrt{5}-\sqrt{\left(\sqrt{5}-1\right)^2}}\)
\(\Leftrightarrow C=\sqrt{\sqrt{5}-\sqrt{5}+1}=\sqrt{1}=1\)
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Bài 1: Tính giá trị của biểu thức:frac{1}{1sqrt{2}+2sqrt{1}}+frac{1}{2sqrt{3}+3sqrt{2}}+frac{1}{3sqrt{4}+4sqrt{3}}+...+frac{1}{2017sqrt{2018}+2018sqrt{2017}}Bài 2: Chứng minh rằng các biểu thức sau có giá trị là số nguyênA left(sqrt{57}+3sqrt{6}+sqrt{38}+6right)left(sqrt{57}-3sqrt{6}-sqrt{38}+6right)B frac{2sqrt{3+sqrt{5-sqrt{13+sqrt{48}}}}}{sqrt{6}+sqrt{2}}
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Bài 1: Tính giá trị của biểu thức:\(\frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{3\sqrt{4}+4\sqrt{3}}+...+\frac{1}{2017\sqrt{2018}+2018\sqrt{2017}}\)
Bài 2: Chứng minh rằng các biểu thức sau có giá trị là số nguyên
A = \(\left(\sqrt{57}+3\sqrt{6}+\sqrt{38}+6\right)\left(\sqrt{57}-3\sqrt{6}-\sqrt{38}+6\right)\)
B = \(\frac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)
Bài 1: Rút gọn biểu thức:Afrac{a^3-3a+left(a^2-1right)sqrt{a^2-4}-2}{a^3-3a+left(a^2-1right)sqrt{a^2-4}+2}left(a2right)Bsqrt{frac{1}{a^2+b^2}+frac{1}{left(a+bright)^2}+sqrt{frac{1}{a^4}+frac{1}{b^4}+frac{1}{left(a^2+b^2right)^2}}}left(abne0right)Bài 2: Tính giá trị của biểu thức:Efrac{1}{1sqrt{2}+2sqrt{1}}+frac{1}{2sqrt{3}+3sqrt{2}}+frac{1}{3sqrt{4}+4sqrt{3}}+...+frac{1}{2017sqrt{2018}+2018sqrt{2017}}Bài 3: Chứng minh rằng các biểu thức sau có gúa trị là số nguyênAleft(sqrt{57}+3sqrt{6}+sqrt{38}...
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Bài 1: Rút gọn biểu thức:
\(A=\frac{a^3-3a+\left(a^2-1\right)\sqrt{a^2-4}-2}{a^3-3a+\left(a^2-1\right)\sqrt{a^2-4}+2}\left(a>2\right)\)
\(B=\sqrt{\frac{1}{a^2+b^2}+\frac{1}{\left(a+b\right)^2}+\sqrt{\frac{1}{a^4}+\frac{1}{b^4}+\frac{1}{\left(a^2+b^2\right)^2}}}\left(ab\ne0\right)\)
Bài 2: Tính giá trị của biểu thức:
\(E=\frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{3\sqrt{4}+4\sqrt{3}}+...+\frac{1}{2017\sqrt{2018}+2018\sqrt{2017}}\)
Bài 3: Chứng minh rằng các biểu thức sau có gúa trị là số nguyên
\(A=\left(\sqrt{57}+3\sqrt{6}+\sqrt{38}+6\right)\left(\sqrt{57}-3\sqrt{6}-\sqrt{38}+6\right)\)
\(B=\frac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)
Tính:
a,\(\sqrt{19-6\sqrt{2}}\)
b,\(\sqrt{21+12\sqrt{3}}\)
c,\(\sqrt{57-40\sqrt{2}}\)
d,\(\sqrt{\left(5-2\sqrt{6}\right)\left(4-2\sqrt{3}\right)}\)
e,\(\sqrt{21+6\sqrt{6}}+\sqrt{21-6\sqrt{6}}\)
g,\(\sqrt{7-4\sqrt{3}}-\sqrt{7+4\sqrt{3}}\)
\(a.\sqrt{19-6\sqrt{2}}=\sqrt{18-2.3\sqrt{2}+1}=3\sqrt{2}-1\)
\(b.\sqrt{21+12\sqrt{3}}=\sqrt{12+2.2\sqrt{3}.3+9}=2\sqrt{3}+3\)
\(c.\sqrt{57-40\sqrt{2}}=\sqrt{32-2.4\sqrt{2}.5+25}=4\sqrt{2}-5\)
\(d.\sqrt{\left(5-2\sqrt{6}\right)\left(4-2\sqrt{3}\right)}=\sqrt{3-2\sqrt{3}.\sqrt{2}+2}.\sqrt{3-2\sqrt{3}+1}=\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-1\right)\) \(e.\sqrt{21+6\sqrt{6}}+\sqrt{21-6\sqrt{6}}=\sqrt{18+2.3\sqrt{2}.\sqrt{3}+3}+\sqrt{18-2.3\sqrt{2}.\sqrt{3}+3}=3\sqrt{2}+\sqrt{3}+3\sqrt{2}-\sqrt{3}=6\sqrt{2}\) \(g.\sqrt{7-4\sqrt{3}}-\sqrt{7+4\sqrt{3}}=\sqrt{4-2.2\sqrt{3}+3}-\sqrt{4+2.2\sqrt{3}+3}=2-\sqrt{3}-2-\sqrt{3}=-2\sqrt{3}\)
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a)
=\(\sqrt{18-2.3\sqrt{2}.1+1}\)
\(=\sqrt{\left(3\sqrt{2}-1\right)^2}\)
\(=3\sqrt{2}-1\)
b)
=\(\sqrt{12+2.2\sqrt{3}.3+9}\)
=\(\sqrt{\left(2\sqrt{3}+3\right)^2}\)
=\(2\sqrt{3}+3\)
c)
=\(\sqrt{25-2.5.4\sqrt{2}+32}\)
=\(\sqrt{\left(5-4\sqrt{2}\right)^2}\)
=\(4\sqrt{2}-5\)
d)
\(=\sqrt{\left(3-2.\sqrt{3}.\sqrt{2}+2\right)\left(3-2\sqrt{3}+1\right)}\\ =\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2\left(\sqrt{3}-1\right)^2}\\ =\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-1\right)\\ =3-\sqrt{3}-\sqrt{6}+\sqrt{2}\)
e)
\(=\sqrt{18+2.3\sqrt{2}.\sqrt{3}+3}+\sqrt{18-2.3\sqrt{2}.\sqrt{3}+3}\\ =\sqrt{\left(3\sqrt{2}+\sqrt{3}\right)^2}+\sqrt{\left(3\sqrt{2}-\sqrt{3}\right)^2}\\ =3\sqrt{2}+\sqrt{3}+3\sqrt{2}-\sqrt{3}\\ =6\sqrt{2}\)
g)
\(=\sqrt{4-2.2.\sqrt{3}+3}-\sqrt{4+2.2.\sqrt{3}+3}\\ =\sqrt{\left(2-\sqrt{3}\right)^2}-\sqrt{\left(2+\sqrt{3}\right)^2}\\ =2-\sqrt{3}-2-\sqrt{3}\\ =-2\sqrt{3}\)
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b, \(\sqrt{21+12\sqrt{3}}=\sqrt{21+2.3.2.\sqrt{3}}=\sqrt{9+2.3.\sqrt{12}+12}\)
\(=\sqrt{\left(3+\sqrt{12}\right)^2}=3+\sqrt{12}\)
\(c,\sqrt{57-40\sqrt{2}}=\sqrt{57-2.4.5.\sqrt{2}}=\sqrt{25-2.5.\sqrt{32}}\)
\(=\sqrt{\left(5-\sqrt{32}\right)^2}=\left|5-\sqrt{32}\right|=5-\sqrt{32}\)
\(d,\sqrt{\left(5-2\sqrt{6}\right)\left(4-2\sqrt{3}\right)}=\sqrt{\left(3-2.\sqrt{2}.\sqrt{3}+2\right)\left(3-2\sqrt{3}+1\right)}\) \(=\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2\left(\sqrt{3}-1\right)^2}=\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-1\right)\)
\(e,A=\sqrt{21+6\sqrt{6}}+\sqrt{21-6\sqrt{6}}\)
Ta có :
\(21+6\sqrt{6}=\dfrac{42+12\sqrt{6}}{2}=\dfrac{36+2.6.\sqrt{6}+6}{2}=\left(\dfrac{6+\sqrt{6}}{\sqrt{2}}\right)^2\) Tương tự : \(21-6\sqrt{6}=\left(\dfrac{6-\sqrt{6}}{\sqrt{2}}\right)^2\)
Do đó :
\(A=\sqrt{\left(\dfrac{6+\sqrt{6}}{\sqrt{2}}\right)^2}+\sqrt{\left(\dfrac{6-\sqrt{6}}{\sqrt{2}}\right)^2}=\dfrac{6+\sqrt{6}}{\sqrt{2}}+\dfrac{6-\sqrt{6}}{\sqrt{2}}=\dfrac{6+\sqrt{6}+6-\sqrt{6}}{\sqrt{2}}\)\(=\dfrac{12}{\sqrt{2}}=\dfrac{12\sqrt{2}}{2}=6\sqrt{2}\)
Phần g làm tương tự như phần e nha bạn :>
Chúc bạn học tốt :>
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A=\(\left(3x^3+3x^2+2\right)^{1998}\) với x=\(\frac{\left(\sqrt{5}+2\right)\sqrt[3]{17\sqrt{5}-38}}{\sqrt{5}+\sqrt{14-6\sqrt{5}}}\)
Cho x=\(\frac{\left(\sqrt{5}+2\right)\cdot\sqrt[3]{17\sqrt{5}-38}}{\sqrt{5}+\sqrt{14-6\sqrt{5}}}\) Tính A=\(\left(3x^3+8x^2+2\right)^{2018}\)
\(x=\dfrac{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}{\sqrt{5}+3-\sqrt{5}}=\dfrac{3}{3}=1\)
\(A=\left(3\cdot1+8\cdot1+2\right)^{2018}=13^{2018}\)
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Cho \(x=\dfrac{\left(\sqrt{5}+2\right)\sqrt[3]{17\sqrt{5}-38}}{\sqrt{5}+\sqrt{14-6\sqrt{5}}}\)
Tính \(A=\left(3x^3+8x^2+2\right)^{1998}\)
Mẫu của x
\(\sqrt{5}+\sqrt{3^2-2.3.\sqrt{5}+5}=\sqrt{5}+\left|3-\sqrt{5}\right|=3\)
Tử của x
\(\left(\sqrt{5}+2\right)\sqrt[3]{17\sqrt{5}-38}=\left(\sqrt{5}+2\right)\sqrt[3]{\left(5\sqrt{5}\right)-3.\left(\sqrt{5}\right)^2.2+3.\sqrt{5}.2^2-2^3}=\left(\sqrt{5}+2\right)\sqrt{\left(\sqrt{5}-2\right)^3}=\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)=5-4=1\)
=> \(x=\dfrac{1}{3}\)
\(A=\left(\dfrac{3}{3^3}+\dfrac{8}{3^2}+2\right)^{1998}=\left(\dfrac{1+8+9}{3^2}\right)^{1998}=2^{1998}\)
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x^3left(sqrt[3]{5+2sqrt{6}}+sqrt[3]{5-2sqrt{6}}right)^3sqrt[3]{5+2sqrt{6}}^3+3sqrt[3]{left(5+2sqrt{6}right)^2}.sqrt[3]{5-2sqrt{6}}+3sqrt[3]{5+2sqrt{6}}.sqrt[3]{left(5-2sqrt{6}right)^2}+sqrt[3]{5-2sqrt{6}}^35+2sqrt{6}+3sqrt[3]{left(5+2sqrt{6}right)left(5-2sqrt{6}right)}.sqrt[3]{5+2sqrt{6}}+3sqrt[3]{left(5+2sqrt{6}right)left(5-2sqrt{6}right)}.sqrt[3]{5-2sqrt{6}}+5-2sqrt{6}5+5+3sqrt[3]{left(25-4.6right)}.sqrt[3]{5+2sqrt{6}}+3sqrt[3]{left(25-4.6right)}.sqrt[3]{5-2sqrt{6}}10+
3sqrt[3]{5+2sqrt{6}}+3sq...
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\(x^3=\left(\sqrt[3]{5+2\sqrt{6}}+\sqrt[3]{5-2\sqrt{6}}\right)^3=\sqrt[3]{5+2\sqrt{6}}^3\)
\(+3\sqrt[3]{\left(5+2\sqrt{6}\right)^2}.\sqrt[3]{5-2\sqrt{6}}+3\sqrt[3]{5+2\sqrt{6}}.\sqrt[3]{\left(5-2\sqrt{6}\right)^2}+\sqrt[3]{5-2\sqrt{6}}^3\)
\(=5+2\sqrt{6}+3\sqrt[3]{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)}.\sqrt[3]{5+2\sqrt{6}}\)
\(+3\sqrt[3]{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)}.\sqrt[3]{5-2\sqrt{6}}+5-2\sqrt{6}\)
\(=5+5+3\sqrt[3]{\left(25-4.6\right)}.\sqrt[3]{5+2\sqrt{6}}+3\sqrt[3]{\left(25-4.6\right)}.\sqrt[3]{5-2\sqrt{6}}\)
\(=10+ 3\sqrt[3]{5+2\sqrt{6}}+3\sqrt[3]{5-2\sqrt{6}}\)
p/s : có bạn hỏi nên mình đăng , các bạn đừng report nhé
tính \(M=\left(3x^3+8x^2+2\right)^4\)
voi \(x=\frac{\left(\sqrt{5}+2\right).\sqrt[3]{17\sqrt{5}-38}}{\sqrt{5}+\sqrt{14-6\sqrt{5}}}\)