a: \(\dfrac{x^3-x}{3x+3}=\dfrac{x\left(x-1\right)\left(x+1\right)}{3\left(x+1\right)}=\dfrac{x\left(x-1\right)}{3}\)

b: \(\dfrac{x^2-4xy+4y^2-4}{2x^2-4xy+4x}\)

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

\(=\dfrac{x-2y-2}{2x}\)

Xét mẫu số của phân số:

\(\dfrac{1}{99}+\dfrac{2}{98}+...+\dfrac{99}{1}\)

\(=\left(\dfrac{1}{99}+1\right)+\left(\dfrac{2}{98}+1\right)+...+\left(\dfrac{98}{2}+1\right)+\left(\dfrac{99}{1}-98\right)\)

\(=\dfrac{100}{99}+\dfrac{100}{98}+...+\dfrac{100}{2}+1\)

\(=100\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{99}+\dfrac{1}{100}\right)\)

Ta thấy mẫu số gấp tử số 100 lần. Vậy phân số đó có giá trị bằng \(\dfrac{1}{100}\)

\(A=100^2-99^2+98^2-97^2+....+2^2-1^2\)

\(=\left(100-99\right).\left(100+99\right)+\left(98-97\right).\left(98+97\right)+....+\left(2-1\right).\left(2+1\right)\)

\(=1+2+....+97+98+99+100=\frac{100.\left(100+1\right)}{2}=5050\)

\(B=3\left(2^2+1\right)\left(2^4+1\right)....\left(2^{64}+1\right)+1=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{64}+1\right)+1\)

\(=\left(2^4-1\right)\left(2^4+1\right)......\left(2^{64}+1\right)+1=\left(2^8-1\right).....\left(2^{64}+1\right)+1\)

Tiếp tục rút gọn như vậy,ta đc \(B=\left(2^{64}-1\right)\left(2^{64}+1\right)=2^{128}-1+1=2^{128}\)

a)M=2¹⁰⁰-2⁹⁹+2⁹⁸-...+2²-2

2²M=2¹⁰²-2¹⁰¹+2¹⁰⁰-...+2^2-2

4M-M=2¹⁰²-2¹⁰¹+2¹⁰⁰-...+2²-2-2¹⁰⁰+2⁹⁹-...-2²+2

3M=2¹⁰²-2¹⁰¹

M=\(\frac{2^{102}-2^{101}}{3}\)

chết lm sai mất òy

(x-1)(x-2)(x+2)-(x-3)\(^3\)

=(x-1)(x\(^2\)-4)-(x-3)\(^3\)

(xy-1)(xy-2)-(xy-2)\(^2\)

=(xy-2)(xy-1-xy+2)

=xy-2

B=1/2²+1/2³+...+1/2¹⁰⁰

2B=1/2¹+1/2²+...1/2⁹⁹

2B-B=(1/2¹+1/2²+...+1/2⁹⁹)-(1/2²+1/2³+...+1/2¹⁰⁰)

B=1/2¹-1/2¹⁰⁰=2⁹⁹/2¹⁰⁰-1/2¹⁰⁰=2⁹⁹-1/2¹⁰⁰