Với mọi \(k\ge2\)  thì \(\frac{2k+\sqrt{k^2-1}}{\sqrt{k-1}+\sqrt{k+1}}=\frac{\left[\left(\sqrt{k-1}\right)^2+\left(\sqrt{k+1}\right)^2+\sqrt{\left(k-1\right)\left(k+1\right)}\right]\left(\sqrt{k+1}-\sqrt{k-1}\right)}{\left(\sqrt{k-1}+\sqrt{k+1}\right)\left(\sqrt{k+1}-\sqrt{k-1}\right)}\)

                                                \(=\frac{\sqrt{\left(k+1\right)^3}-\sqrt{\left(k-1\right)^3}}{2}\)

Suy ra tổng đã cho có thể viết là :

\(A=\frac{1}{2}\left[\sqrt{3^3}-\sqrt{1^3}+\sqrt{4^3}-\sqrt{2^3}+\sqrt{5^3}-\sqrt{3^3}+\sqrt{6^3}-\sqrt{4^3}+...+\sqrt{101^3}-\sqrt{99^3}\right]\)

    \(=\frac{1}{2}\left[-1-\sqrt{2^3}+\sqrt{101^3}+\sqrt{100^3}\right]\)

   \(=\frac{999+\sqrt{101^3}-\sqrt{8}}{2}\)

\(a.a\ne\pm1\)

\(b.K=\dfrac{1}{a+1}+\dfrac{2}{a^2-1}=\dfrac{a-1}{\left(a-1\right)\left(a+1\right)}+\dfrac{2}{\left(a-1\right)\left(a+1\right)}=\dfrac{a+1}{\left(a-1\right)\left(a+1\right)}=\dfrac{1}{a-1}\)

\(c.K=\dfrac{1}{1-\dfrac{1}{2}}=\dfrac{1}{\dfrac{1}{2}}=2\)

a: ĐKXĐ: x<>1; x<>2; x<>3

\(K=\left(\dfrac{x^2}{\left(x-2\right)\left(x-3\right)}+\dfrac{x^2}{\left(x-1\right)\left(x-2\right)}\right)\cdot\dfrac{\left(x-1\right)\left(x-3\right)}{x^4+2x^2+1-x^2}\)

\(=\dfrac{x^3-x^2+x^3-3x^2}{\left(x-2\right)\left(x-3\right)\left(x-1\right)}\cdot\dfrac{\left(x-1\right)\left(x-3\right)}{\left(x^2+1+x\right)\left(x^2+1-x\right)}\)

\(=\dfrac{2x^3-4x^2}{\left(x-2\right)}\cdot\dfrac{1}{\left(x^2+x+1\right)\left(x^2-x+1\right)}\)

\(=\dfrac{2x^2\left(x-2\right)}{\left(x-2\right)\left(x^4+x^2+1\right)}=\dfrac{2x^2}{x^4+x^2+1}\)

b:

90.10^k-10^k+2+10^k+1

=9.10.10^k-10^k+1.10+10^k+1

=10^k+1.(9-10+1)

=10^k+1.0

=0

\(90.10^k-10^{k+2}+10^{k+1}\)

\(=10^k\left(90-10^2+10\right)\)

\(=10^k.0=0\)

= 0

mik mới khôi phục đc điểm âm mà giờ lại bị trừ tiếp, các bạn giúp mik nha, mik sẽ giúp lại 100%

cảm mơn các bạn trc ~~~

\(K=\left(\frac{a}{a-1}-\frac{1}{a^2-a}\right):\left(\frac{1}{a+1}+\frac{2}{a^2+1}\right)\)

a/ K xác định khi \(\hept{\begin{cases}a-1\ne0\\a^2-a=a\left(a-1\right)\ne0\\a+1\ne0\end{cases}}\) <=> \(\hept{\begin{cases}a\ne\pm1\\a\ne0\end{cases}}\)

b/ \(K=\left(\frac{a}{a-1}-\frac{1}{a^2-a}\right):\left(\frac{1}{a+1}+\frac{2}{a^2+1}\right)=\left(\frac{a}{a-1}-\frac{1}{a\left(a-1\right)}\right):\left(\frac{1}{a+1}+\frac{2}{a^2+1}\right)\)

=> \(K=\frac{a^2-1}{a\left(a-1\right)}:\frac{a^2+1+2a+2}{\left(a+1\right)\left(a^2+1\right)}\)

=> \(K=\frac{\left(a-1\right)\left(a+1\right)}{a\left(a-1\right)}.\frac{\left(a+1\right)\left(a^2+1\right)}{a^2+2a+3}\)

=> \(K=\frac{\left(a+1\right)^2\left(a^2+1\right)}{a\left(a^2+2a+3\right)}\)

c/ a=1/2 

=> \(K=\frac{\left(\frac{1}{2}+1\right)^2\left(\frac{1}{4}+1\right)}{\frac{1}{2}\left(\frac{1}{4}+1+3\right)}=\frac{\frac{9}{4}.\frac{5}{4}}{\frac{17}{8}}=\frac{45}{16}.\frac{8}{17}=\frac{45}{2.17}\)

=> \(K=\frac{45}{34}\)

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

ở phàn a+/a thiếu số 1 nhé

\(\frac{1}{a+1}+\frac{2}{a^2-1}=\frac{a-1+2}{a^2-1}=\frac{1}{a-1}\)

=> K =\(\frac{a^2-1}{a}\) 

đkxđ: a khác +-1

b, thay vào mà tình

a/ \(K=\left(\frac{a}{a-1}-\frac{1}{a^2-a}\right):\left(\frac{1}{a+1}+\frac{2}{a^2-1}\right)\)

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

\(=\frac{a^2-1}{a\left(a-1\right)}:\frac{a-1+2}{\left(a-1\right)\left(a+1\right)}\)

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

\(=\frac{a+1}{a}.a+1\)

\(=\frac{\left(a+1\right)^2}{a}\)

b, Thay a=1/2

\(\Rightarrow\frac{\left(\frac{1}{2}+1\right)^2}{\frac{1}{2}}=\frac{\frac{9}{4}}{\frac{1}{2}}=\frac{9}{2}\)