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22 tháng 10 2016

ta có: A3=\(6\sqrt{3}+10-6\sqrt{3}+10-3\sqrt[3]{\left(6\sqrt{3}+10\right)\left(6\sqrt{3}-10\right)}.\left(\sqrt[3]{6\sqrt{3}+10}-\sqrt[3]{6\sqrt{3}-10}\right)\)

=\(20-3.\sqrt[3]{8}.A\)=\(20-6A\)

do đó A3=20-6A↔A3+6A-20=0↔(A2+2A+10)(A-2)=0

dễ thấy A2+2A+10>0→A=2

b) giống a)

c)giống b)

10 tháng 8 2017

1)

dat \(a=\sqrt[3]{x+1};b=\sqrt[3]{7-x}\)

ta co b=2-a

a^3+b^3=x+1+7-x=8 

a^3+b^3=a^3+b^3+3ab(a+b)

ab(a+b)=0

suy ra a=0 hoac b=0 hoac a=-b

<=> x=-1; x=7 

a=-b

a^3=-b^3

x+1=x+7 (vo li nen vo nghiem)

cau B tuong tu

2)

tat ca cac bai tap deu chung 1 dang do la

\(\sqrt[3]{a+m}+\sqrt[3]{b-m}\)voi m la tham so

dang nay co 2 cach 

C1 lap phuong VD: \(B^3=10+3\sqrt[3]{< 5+2\sqrt{13}>< 5-2\sqrt{13}>}\left(B\right)\)

B^3=10-9B

B=1 cach nay nhanh nhung kho nhin

C2 dat an

\(a=\sqrt[3]{5+2\sqrt{13}};b=\sqrt[3]{5-2\sqrt{13}}\)

de thay B=a+b

a^3+b^3=10

ab=-3

B^3=10-9B

suy ra B=1

tuong tu giai cac cau con lai.

10 tháng 8 2017

Bài 1:

a. Đặt \(a=\sqrt[3]{x+1}\)\(b=\sqrt[3]{7-x}\). Ta có:

\(\hept{\begin{cases}a+b=2\\a^3+b^3=8\end{cases}\Leftrightarrow a^3+\left(2-a\right)^3=8\Leftrightarrow...\Leftrightarrow\orbr{\begin{cases}a=0\\a=2\end{cases}}}\)

\(\Leftrightarrow\hept{\begin{cases}a=0\\b=2\end{cases}}\)hoặc \(\hept{\begin{cases}a=2\\b=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}\sqrt[3]{x+1}=0\\\sqrt[3]{7-x}=2\end{cases}}\)hoặc \(\hept{\begin{cases}\sqrt[3]{x+1}=2\\\sqrt[3]{7-x}=0\end{cases}}\)

\(\Leftrightarrow x=-1\)hoặc \(x=7\)

3 tháng 7 2017

\(P=\dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+...+\dfrac{1}{\sqrt{99}+\sqrt{100}}\)

\(P=\dfrac{\sqrt{2}-1}{\left(1+\sqrt{2}\right)\left(\sqrt{2}-1\right)}+\dfrac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{3}-\sqrt{2}\right)}+...+\dfrac{\sqrt{100}-\sqrt{99}}{\left(\sqrt{99}+\sqrt{100}\right)\left(\sqrt{100}-\sqrt{99}\right)}\)

\(P=\dfrac{\sqrt{2}-1}{2-1}+\dfrac{\sqrt{3}-\sqrt{2}}{3-2}+...+\dfrac{\sqrt{100}-\sqrt{99}}{100-99}\)

\(P=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{100}-\sqrt{99}\)

\(P=-1+\sqrt{100}=-1+10=9\)

3 tháng 7 2017

Áp dụng:\(\dfrac{1}{\sqrt{a}+\sqrt{a+1}}=\dfrac{\sqrt{a+1}-\sqrt{a}}{\left(\sqrt{a}+\sqrt{a+1}\right)\left(\sqrt{a+1}-\sqrt{a}\right)}=\dfrac{\sqrt{a+1}-\sqrt{a}}{a+1-a}=\sqrt{a+1}-\sqrt{a}\)

9 tháng 7 2017

ĐKXĐ: x>0

\(\left(\dfrac{x}{x+3\sqrt{x}}+\dfrac{1}{\sqrt{x}+3}\right):\left(1-\dfrac{2}{\sqrt{x}}+\dfrac{6}{x+3\sqrt{x}}\right)\)

= \(\dfrac{x+\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+3\right)}:\dfrac{x+3\sqrt{x}-2\left(\sqrt{x}+3\right)+6}{\sqrt{x}\left(\sqrt{x}+3\right)}\)

= \(\dfrac{x+\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+3\right)}:\dfrac{x+\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+3\right)}=1\)

9 tháng 7 2017

cảm ơn bạn nhiều!!!

a: \(D=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\cdot\sqrt{8-2\sqrt{15}}\)

\(=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)\)

\(=32-8\sqrt{15}+8\sqrt{15}-30=2\)

b: \(E=\sqrt{6-2\sqrt{5}}\cdot\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)\)

\(=\left(6-2\sqrt{5}\right)\left(3+\sqrt{5}\right)\)

\(=18+6\sqrt{5}-6\sqrt{5}-10=8\)

\(=\sqrt{2}\left(\dfrac{2+\sqrt{5}}{2+\sqrt{5}+1}+\dfrac{2-\sqrt{5}}{2-\sqrt{5}+1}\right)\)

\(=\sqrt{2}\left(\dfrac{\left(2+\sqrt{5}\right)\left(3-\sqrt{5}\right)+\left(2-\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}\right)\)

\(=\sqrt{2}\cdot\dfrac{6-2\sqrt{5}+3\sqrt{5}-5+6+2\sqrt{5}-3\sqrt{5}-5}{4}\)

\(=\sqrt{2}\cdot\dfrac{2}{4}=\dfrac{\sqrt{2}}{2}\)

11 tháng 8 2017

a/ \(\sqrt{8-2\sqrt{15}}-\sqrt{8+2\sqrt{15}}=\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}\) \(=\sqrt{5}-\sqrt{3}-\sqrt{5}-\sqrt{3}=-2\sqrt{3}\).

b/ \(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\Rightarrow A^2=8+2\sqrt{4^2-\left(\sqrt{10+2\sqrt{5}}\right)^2}=8+2\sqrt{6-2\sqrt{5}}\) \(=8+2\sqrt{\left(\sqrt{5}-1\right)^2}=8+2\sqrt{5}-2=6+2\sqrt{5}=\left(\sqrt{5}+1\right)^2\)

\(\Rightarrow A=\sqrt{5}+1\)

c/ \(B=\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-\sqrt{5}}\Rightarrow\sqrt{2}B=\sqrt{8+2\sqrt{15}}+\sqrt{8-2\sqrt{15}}-2\sqrt{6-2\sqrt{5}}=\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}-2\sqrt{\left(\sqrt{5}-1\right)^2}=\sqrt{5}+\sqrt{3}+\sqrt{5}-\sqrt{3}-2\sqrt{5}+2=2\Rightarrow B=\sqrt{2}\)

21 tháng 10 2016

A = \(\sqrt[3]{10+6\sqrt{3}}+\sqrt[3]{10-6\sqrt{3}}\)

<=> A3 = 20 - 3×2A

<=> A3 + 6A - 20 = 0

<=> A = 2

21 tháng 10 2016

2 câu còn lại làm tương tự 

12 tháng 7 2017

1a) \(\sqrt{4+\sqrt{8}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}\)

\(=\sqrt{4+\sqrt{8}}.\sqrt{\left(2+\sqrt{2+\sqrt{2}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{2}}}\right)}\)

\(=\sqrt{4+\sqrt{8}}.\sqrt{4-2-\sqrt{2}}\)

\(=\sqrt{4+\sqrt{8}}.\sqrt{2-\sqrt{2}}=\sqrt{\left(4+\sqrt{8}\right)\left(2-\sqrt{2}\right)}\)

\(=\sqrt{8-4\sqrt{2}-\sqrt{16}+2\sqrt{8}}\)

\(=\sqrt{8-4\sqrt{2}-4+4\sqrt{2}}\)

\(=\sqrt{4}=2\)

12 tháng 7 2017

1b) \(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)

\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{4+4\sqrt{3}+3}}}\)

\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}\)

\(=\sqrt{5\sqrt{3}+5\sqrt{48-20-10\sqrt{3}}}\)

\(=\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}\)

\(=\sqrt{5\sqrt{3}+5\sqrt{25-10\sqrt{3}+3}}\)

\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)

\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}\)

\(=\sqrt{25}=5\)

2 tháng 7 2016

\(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}=\frac{\left(\sqrt[3]{3+2\sqrt{2}}\right)^3+\left(\sqrt[3]{3-2\sqrt{2}}\right)^3}{\left(\sqrt{3+2\sqrt{2}}\right)^2-\sqrt{\left(3-2\sqrt{2}\right)\left(3+2\sqrt{2}\right)}+\left(\sqrt{3-2\sqrt{2}}\right)^2}\)

\(\frac{6}{3+2\sqrt{2}+3-2\sqrt{2}+\sqrt{9-8}}=\frac{6}{6+1}=\frac{6}{7}\)

2 tháng 7 2016

cho mình hỏi làm thế nào để ra biểu thức thứ 2 vậy