\(BT=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\sqrt{8-2\sqrt{15}}=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)\)

\(=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)=\frac{8^2-60}{2}=2\)

1.\(\frac{5+\sqrt{5}}{5-\sqrt{5}}+\frac{5-\sqrt{5}}{5+\sqrt{5}}=\frac{\left(5+\sqrt{5}\right)\left(5+\sqrt{5}\right)}{\left(5-\sqrt{5}\right)\left(5+\sqrt{5}\right)}+\frac{\left(5-\sqrt{5}\right)\left(5-\sqrt{5}\right)}{\left(5-\sqrt{5}\right)\left(5+\sqrt{5}\right)}\)

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

\(=\frac{25+10\sqrt{5}+5+25-10\sqrt{5}+5}{20}\)

\(=\frac{60}{20}=3\)

2.

a) \(\sqrt{45x}-2\sqrt{20x}+2\sqrt{80x}=21\)

ĐK : x ≥ 0

<=> \(\sqrt{5x\cdot9}-2\sqrt{5x\cdot4}+2\sqrt{5x\cdot16}=21\)

<=> \(\sqrt{5x\cdot3^2}-2\sqrt{2^2\cdot5x}+2\sqrt{5x\cdot4^2}=21\)

<=> \(\left|3\right|\sqrt{5x}-2\cdot\left|2\right|\sqrt{5x}+2\cdot\left|4\right|\sqrt{5x}=21\)

<=> \(\sqrt{5x}\cdot\left(3-4+8\right)=21\)

<=> \(\sqrt{5x}\cdot7=21\)

<=> \(\sqrt{5x}=3\)

<=> \(5x=9\)

<=> \(x=\frac{9}{5}\left(tm\right)\)

ơ đang làm lại bấm " Gửi trả lời " ._.

2b) \(\sqrt{x^2-10x+25}=4\)

<=> \(\sqrt{\left(x-5\right)^2}=4\)

<=> \(\left|x-5\right|=4\)

<=> \(\orbr{\begin{cases}x-5=4\\x-5=-4\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=9\\x=1\end{cases}}\)

3. \(A=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right)\div\left(\frac{\sqrt{x}+1}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)

ĐK : \(\hept{\begin{cases}x>0\\x\ne1\\x\ne4\end{cases}}\)

\(=\left(\frac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x-1}\right)}\right)\div\left(\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\left(\frac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right)\div\left(\frac{x-1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\frac{x-4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\div\left(\frac{x-1-x+4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\div\frac{3}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\times\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}{3}\)

\(=\frac{\sqrt{x}-2}{3\sqrt{x}}\)

Bài 1: Ta có: \(\sqrt{2020}-\sqrt{2019}=\frac{1}{\sqrt{2020}+\sqrt{2019}};\)\(\sqrt{2018}-\sqrt{2017}=\frac{1}{\sqrt{2018}+\sqrt{2017}}\)

Dễ thấy \(\sqrt{2020}+\sqrt{2019}>\sqrt{2018}+\sqrt{2017}\)nên \(\frac{1}{\sqrt{2020}+\sqrt{2019}}< \frac{1}{\sqrt{2018}+\sqrt{2017}}\)

Suy ra\(\sqrt{2020}-\sqrt{2019}< \sqrt{2018}-\sqrt{2017}\)

Bài 2: Xét biểu thức \(\sqrt{a^2+a^2\left(a+1\right)^2+\left(a+1\right)^2}=\sqrt{a^2\left(a^2+2a+1+1\right)+\left(a+1\right)^2}=\sqrt{a^4+2a^2\left(a+1\right)+\left(a+1\right)^2}=\sqrt{\left(a^2+a+1\right)^2}=a^2+a+1\)(Vì \(a^2+a+1>0\forall a\inℝ\))

Áp dụng công thức tổng quát trên, ta được: \(\sqrt{2019^2+2019^2.2020^2+2020^2}=2019^2+2019+1\)(là số tự nhiên) (đpcm)

Dạng tổng quát ta càn chứng minh \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\)

Ta có \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}\)

\(=\sqrt{\frac{a^4+2a^3b+a^2b^2+2ab^3+b^4}{a^2b^2\left(a+b\right)^2}}\)

\(=\sqrt{\left(\frac{a^2+ab+b^2}{ab\left(a+b\right)}\right)^2}\)

\(=\frac{a^2+ab+b^2}{ab\left(a+b\right)}=\frac{1}{b}+\frac{b}{a\left(a+b\right)}=\frac{1}{b}+\frac{1}{a}-\frac{1}{a+b}\left(đpcm\right)\)

Áp dụng dạng trên ta được 

\(D=1+\frac{1}{1}-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+1+\frac{1}{3}-\frac{1}{4}+...+1+\frac{1}{99}-\frac{1}{100}\)

\(D=100-\frac{1}{100}=\frac{9999}{100}\)

Xét biểu thức \(A=\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}\)với a > 0

\(A^2=1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}=\frac{a^2\left(a+1\right)^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}=\frac{a^2\left(a^2+2a+1+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}=\frac{a^4+2a^2\left(a+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}=\frac{\left(a^2+a+1\right)^2}{a^2\left(a+1\right)^2}=\left[\frac{a^2+a+1}{a\left(a+1\right)}\right]^2\)Do a > 0 nên A > 0 và \(A=\frac{a^2+a+1}{a\left(a+1\right)}=1+\frac{1}{a\left(a+1\right)}=1+\frac{1}{a}-\frac{1}{a+1}\)

Do đó \(D=\left(1+\frac{1}{1}-\frac{1}{2}\right)+\left(1+\frac{1}{2}-\frac{1}{3}\right)+\left(1+\frac{1}{3}-\frac{1}{4}\right)+...+\left(1+\frac{1}{99}-\frac{1}{100}\right)=99+\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\right)=100-\frac{1}{100}=99,99\)

Dạng tổng quát: Với n là các số lẻ lớn hơn hoặc bằng 3 thì \(\frac{1}{n\sqrt{n-2}+\left(n-2\right)\sqrt{n}}=\frac{1}{\sqrt{n\left(n-2\right)}\left(\sqrt{n}+\sqrt{n-2}\right)}=\frac{1}{\sqrt{n\left(n-2\right)}.\frac{2}{\sqrt{n}-\sqrt{n-2}}}=\frac{\sqrt{n}-\sqrt{n-2}}{2\sqrt{n\left(n-2\right)}}=\frac{1}{2}\left(\frac{1}{\sqrt{n-2}}-\frac{1}{\sqrt{n}}\right)\)Áp dụng, ta được: \(C=\frac{1}{3\sqrt{1}+1\sqrt{3}}+\frac{1}{5\sqrt{3}+3\sqrt{5}}+...+\frac{1}{121\sqrt{119}+119\sqrt{121}}=\frac{1}{2}\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{5}}+...+\frac{1}{\sqrt{119}}-\frac{1}{\sqrt{121}}\right)=\frac{1}{2}\left(1-\frac{1}{11}\right)=\frac{5}{11}\)Vậy C = ⁵/₁₁

Xét :\(\frac{1}{\left(a+2\right)\sqrt{a}+a\sqrt{a+2}}=\frac{1}{\sqrt{a}.\sqrt{a+2}\left(\sqrt{a+2}+\sqrt{a}\right)}=\frac{\sqrt{a+2}-\sqrt{a}}{2\sqrt{a}.\sqrt{a+2}}=\frac{1}{2\sqrt{a}}-\frac{1}{2\sqrt{a+2}}\)

Xét: 

\(C=\frac{1}{3\sqrt{1}+1\sqrt{3}}+\frac{1}{5\sqrt{3}+3\sqrt{5}}+...+\frac{1}{121\sqrt{119}+119\sqrt{121}}\)

\(=\frac{1}{2}-\frac{1}{2\sqrt{3}}+\frac{1}{2\sqrt{3}}-\frac{1}{2\sqrt{5}}+\frac{1}{2\sqrt{5}}-\frac{1}{2\sqrt{7}}+...+\frac{1}{2\sqrt{119}}-\frac{1}{2\sqrt{121}}\)

\(=\frac{1}{2}-\frac{1}{2\sqrt{121}}=\frac{1}{2}-\frac{1}{2.11}=\frac{5}{11}\)

Chứng minh với mọi số nguyên dương, ta có:

\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\text{[}\left(n+1\right)\sqrt{n}\text{]}^2-\left(n\sqrt{n+1}\right)^2}\)\(=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\text{ }\left(n+1\right)^2.n-n^2.\left(n+1\right)}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)n\left(n+1-n\right)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)

Áp dụng: Tính B=....

\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\left(\frac{-1}{\sqrt{120}}\right)+\frac{1}{\sqrt{120}}-\frac{1}{\sqrt{121}}=1-\frac{1}{11}=\frac{10}{11}\)

a) \(ĐKXĐ:x\ne1;x\ge0\)

b) Với \(x\ne1;x\ge0\)thì \(A=\left(\frac{1}{\sqrt{x}-1}-\frac{\sqrt{x}}{x-1}\right):\frac{1}{\sqrt{x}+1}=\left(\frac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}}{x-1}\right):\frac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{1}{x-1}.\frac{x-1}{\sqrt{x}-1}=\frac{1}{\sqrt{x}-1}\)c) \(A=-\frac{1}{2}\Leftrightarrow\frac{1}{\sqrt{x}-1}=-\frac{1}{2}\Leftrightarrow2=1-\sqrt{x}\Leftrightarrow\sqrt{x}=-1\)(loại vì \(\sqrt{x}\ge0\forall x\inℕ\))

Vậy không tồn tại giá trị của x để A = ¹/₂

Biến đổi: \(x^2+2xy+y^2=xy+x-y-1\)

\(\Leftrightarrow2x^2+2xy+2y^2-2x+2y+2=0\)

\(\Leftrightarrow\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2=0\Leftrightarrow x=1;y=-1\)

a) đk: \(\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)

b) Ta có:

\(A=\left(\frac{1}{\sqrt{x}-1}-\frac{\sqrt{x}}{x-1}\right)\div\frac{1}{\sqrt{x}+1}\)

\(A=\frac{\sqrt{x}+1-\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\left(\sqrt{x}+1\right)\)

\(A=\frac{1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\left(\sqrt{x}+1\right)\)

\(A=\frac{1}{\sqrt{x}-1}\)

c) Ta có; \(A=-\frac{1}{2}\)

\(\Leftrightarrow\frac{1}{\sqrt{x}-1}=-\frac{1}{2}\)

\(\Leftrightarrow\sqrt{x}-1=-2\)

\(\Leftrightarrow\sqrt{x}=-1\) (vô lý)

Vậy không tồn tại x để A = -1/2