Thực hiện các phép tính sau:
a) x 6 + 2 x 3 + 3 x 3 − 1 . 3 x x + 1 . x 2 + x + 1 x 6 + 2 x 3 + 3 với x ≠ ± 1 ;
b) a 3 + 2 a 2 − a − 2 3 a + 15 . 1 a − 1 − 2 a + 1 + 1 a + 2 với a ≠ − 5 ; − 2 ; ± 1 .
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a: \(\dfrac{6}{x^2+4x}+\dfrac{3}{2x+8}\)
\(=\dfrac{6}{x\left(x+4\right)}+\dfrac{3}{2\left(x+4\right)}\)
\(=\dfrac{12+3x}{2x\left(x+4\right)}=\dfrac{3\left(x+4\right)}{2x\left(x+4\right)}=\dfrac{3}{2x}\)
b: \(\dfrac{x+1}{2x-2}+\dfrac{x-1}{2x+2}+\dfrac{x^2}{1-x^2}\)
\(=\dfrac{x+1}{2\left(x-1\right)}+\dfrac{x-1}{2\left(x+1\right)}-\dfrac{x^2}{\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{\left(x+1\right)^2+\left(x-1\right)^2-2x^2}{2\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{x^2+2x+1+x^2-2x+1-2x^2}{2\left(x-1\right)\left(x+1\right)}=\dfrac{2}{2\left(x-1\right)\left(x+1\right)}=\dfrac{1}{x^2-1}\)
c: \(\dfrac{1}{x^2+xy}+\dfrac{2}{y^2-x^2}+\dfrac{1}{xy-x^2}\)
\(=\dfrac{1}{x\left(x+y\right)}-\dfrac{2}{\left(x-y\right)\left(x+y\right)}-\dfrac{1}{x\left(x-y\right)}\)
\(=\dfrac{x-y-2x-x-y}{x\left(x-y\right)\left(x+y\right)}=\dfrac{-2x-2y}{x\left(x-y\right)\left(x+y\right)}\)
\(=-\dfrac{2}{x\left(x-y\right)}\)
a/ \(\left(2x+3\right)\left(x-5\right)-\left(x-1\right)^2+x\left(7-x\right)\)
\(=2x^2-2x-15-x^2+2x-1+7x-x^2\)
\(=7x-16\)
\(\left(\dfrac{x}{x+1}+\dfrac{x-1}{x}\right):\left(\dfrac{x}{x+1}-\dfrac{x-1}{x}\right)\) \(\left(đk:x\ne0;-1\right)\)
\(=\dfrac{x^2+\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}:\left(\dfrac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}\right)\)
\(=\dfrac{x^2+x^2-1}{x\left(x+1\right)}.\dfrac{x\left(x+1\right)}{x^2-x^2+1}\)
\(=\dfrac{\left(2x^2-1\right)x\left(x+1\right)}{x\left(x+1\right)}=2x^2-1\)
a) \(18x^4y^3:12\left(-x\right)^3y\)
\(=\left(18:-12\right)\left(x^4:x^3\right)\left(y^3:y\right)\)
\(=-\dfrac{3}{2}xy^2\)
b) \(x^2y^2-2xy^3:\dfrac{1}{2}xy^2\)
\(=\dfrac{xy^2\left(x-2y\right)}{\dfrac{1}{2}xy^2}\)
\(=\dfrac{x-2y}{\dfrac{1}{2}}\)
\(=2x-4y\)
\(a,\dfrac{8y}{3x^2}.\dfrac{9x^2}{4y^2}=\dfrac{72x^2y}{12x^2y^2}=\dfrac{6}{y}\\b,\dfrac{3x+x^2}{x^2+x+1}.\dfrac{3x^3-3}{x+3}=\dfrac{x\left(x+3\right)3\left(x-1\right)\left(x^2+x+1\right)}{\left(x^2+x+1\right)\left(x+3\right)}=3x\left(x-1\right)=3x^2-3x \)
\(c,\dfrac{2x^2+4}{x-3}.\dfrac{3x+1}{x-1}.\dfrac{6-2x}{x^2+2}=\dfrac{2\left(x^2+2\right)\left(3x+1\right)2\left(3-x\right)}{\left(x-3\right)\left(x-1\right)\left(x^2+2\right)}=\dfrac{-4\left(3x+1\right)}{x-1}=\dfrac{-12x-4}{x-1}\)
\(d,\dfrac{2x^2}{3y^3}:\left(-\dfrac{4x^3}{21y^2}\right)=\dfrac{-2x^2.21y^2}{3y^3.4x^3}=\dfrac{-42x^2y^2}{12x^3y^3}=\dfrac{-7}{2xy}\)
\(e,\dfrac{2x+10}{x^3-64}:\dfrac{\left(x+5\right)^2}{2x-8}=\dfrac{2\left(x+5\right)}{\left(x-4\right)\left(x^2+4x+16\right)}.\dfrac{2\left(x-4\right)}{\left(x+5\right)^2}=\dfrac{4}{\left(x+5\right)\left(x^2+4x+16\right)}=\dfrac{4}{x^3+9x^2+16x+80}\)
\(f,\dfrac{1}{x+y}\left(\dfrac{x+y}{xy}-x-y\right)-\dfrac{1}{x^2}:\dfrac{y}{x}=\dfrac{1}{x+y}\left(\dfrac{\left(x+y\right)\left(1-xy\right)}{xy}\right)-\dfrac{x}{x^2y}=\dfrac{1-xy}{xy}-\dfrac{x}{x^2y}=\dfrac{-x^2y}{x^2y}=-1\)
\(a,\dfrac{x^2-9}{x-2}:\dfrac{x-3}{x}\\ =\dfrac{\left(x-3\right)\left(x+3\right)}{x-2}\times\dfrac{x}{x-3}\\ =\dfrac{x\left(x+3\right)}{\left(x-2\right)}\)
\(b,\dfrac{x}{z^2}.\dfrac{xz}{y^3}:\dfrac{x^3}{yz}\\ =\dfrac{x}{z^2}.\dfrac{xz}{y^3}.\dfrac{yz}{x^3}=\dfrac{x^2yz^2}{z^2y^3x^3}=\dfrac{1}{xy^2}\)
\(c,\dfrac{2}{x}-\dfrac{2}{x}:\dfrac{1}{x}+\dfrac{4}{x}.\dfrac{x^2}{2}\\ =\dfrac{2}{x}-\dfrac{2}{x}\times\dfrac{x}{1}+\dfrac{4x^2}{2x}\\ =\dfrac{2}{x}-\dfrac{2}{1}+2x\\ =\dfrac{2-2x+2x^2}{x}\)
a) \(\dfrac{x^2-9}{x-2}:\dfrac{x-3}{x}\)
\(=\dfrac{\left(x+3\right)\left(x-3\right)}{x-2}\cdot\dfrac{x}{x-3}\)
\(=\dfrac{x\left(x+3\right)}{x-2}\)
b) \(\dfrac{x}{z^2}\cdot\dfrac{xz}{y^3}:\dfrac{x^3}{yz}\)
\(=\dfrac{x}{z^2}\cdot\dfrac{xz}{y^3}\cdot\dfrac{yz}{x^3}\)
\(=\dfrac{1}{xy^2}\)
c) \(\dfrac{2}{x}-\dfrac{2}{x}:\dfrac{1}{x}+\dfrac{4}{x}\cdot\dfrac{x^2}{2}\)
\(=\dfrac{2}{x}-\dfrac{2}{x}\cdot x+\dfrac{4}{x}\cdot\dfrac{x^2}{2}\)
\(=\dfrac{2}{x}\cdot\left(1-x+2\right)\)
\(=\dfrac{2}{x}\cdot\left(3-x\right)\)
\(=\dfrac{6}{x}-2\)
1.
a) 503 + 120 = 623
b) 1000 - 120 = 880
c) 2 + 18 : 2 = 2 + 9 = 11
d) 21 : 7 - 3 = 3 - 3 = 0
2.
a) x - 3 = 21 => x = 24
b) 15 - x . 3 = 6 => 15 - 3x = 6 => 3x = 15 - 6 = 9 => x = 3
c) x + 21 : 7 = 6 => x + 3 = 6 => x = 3
d) 44 + x : 3 = 50 => x : 3 = 50 - 44 = 6 => x = 18
3.
a) 15 . (21 - 3 . 7) = 15 . (21 - 21) = 15 . 0 = 0
b) (4 : 2 - 2) . 105 = (2 - 2) . 105 = 0 . 105 = 0
c) 376 + 285 + 124 + 715 = 1500
d) 97 + 998 + 9999 + 16 = 11110
e) 252 + 139 - 52 - 39 = 300
4.
a) b - a = 5 - 3 = 2
b) a + b = 3 + 5 = 8
c) 2a + b = 2*3 + 5 = 6 + 5 = 11
d) a . (b + 1) = 3 . (5 + 1) = 3 . 6 = 18
5.
a) 37581 - 9999 = 27582
b) 7345 - 1998 = 5347
c) 485321 - 99999 = 385322
d) 7593 - 1997 = 5596
6.
a) (x - 42) - 110 = 0 => x - 42 = 110 => x = 110 + 42 = 152
b) 2436 : x = 12 => x = 2436 / 12 = 203
c) 74 . (x - 3) = 0 => x - 3 = 0 => x = 3
d) x - 36 : 18 = 2 => x - 2 = 36 => x = 36 + 2 = 38
7.
a) 67 + 135 + 33 = 235
b) 997 + 86 = 1083
c) 37 . 38 + 62 . 37 = 1406
d) 43 . 11 = 473
e) 67 . 99 = 6633
8.
a) 71 - (33 + x) = 26 => 71 - 33 - x = 26 => 38 - x = 26 => x = 38 - 26 = 12
b) 97 - (64 - x) = 44 => 97 - 64 + x = 44 => x = 44 - 97 + 64 => x = 11
c) x - 36 : 18 = 12 => x - 2 = 12 => x = 14
d) 3636 : (12 . x - 91) = 36 => 3636 = 36 * (12 . x - 91) => 3636 = 432 . x - 3276 => 432 . x = 3636 + 3276 => 432 . x = 6912 => x = 6912 / 432 => x = 16
e) ( x : 23 + 45) . 67 = 8911 => (x / 23 + 45) . 67 = 8911 => (x / 23 + 45) = 8911 / 67 => (x / 23 + 45) = 133 => x / 23 = 133 - 45 => x / 23 = 88 => x = 88 . 23 => x = 2024
9.
a) 1 + 2 + 3 + ... + 1998 + 1999 = (1999 . (1999 + 1)) / 2 = 1999 . 2000 / 2 = 1999 . 1000 = 1,999,000
b) Tính tổng tất cả các số tự nhiên có 3 chữ số: Tổng các số từ 100 đến 999 = (100 + 999) / 2 * (999 - 100 + 1) = 1099 / 2 * 900 = 549.5 * 900 = 494550
c) Tính tổng tất cả các số lẻ có 3 chữ số: Các số lẻ từ 101 đến 999 là 101, 103, 105, ..., 999. Số lượng các số này là 450 (900 / 2). Tổng các số này là (101 + 999) / 2 * (450) = 550 * 450 = 247,500
10.
a) 53 . 39 + 47 . 39 - 53 . 21 - 47 . 21 = 2079 + 1833 - 1113 - 987 = 2912
b) 2 . 53 . 12 + 4 . 6 . 87 - 3 . 8 . 40 = 1272 + 1044 - 960 = 1356
c) 47 . 29 - 13 . 29 - 14 . 29 = 1363 - 377 - 406 = 580
d) 1754 : 17 - 74 : 17 + 20 : 17 = 103 - 4 + 1 = 100
e) 26 . 7 - 17 . 9 + 13 . 26 - 17 . 11 = 182 - 153 + 338 - 187 = 180
1.
\(\sqrt{50}-3\sqrt{8}+\sqrt{32}=5\sqrt{2}-6\sqrt{2}+4\sqrt{2}=3\sqrt{2}\)
2.
a, ĐK: \(x\in R\)
\(pt\Leftrightarrow\sqrt{\left(x-2\right)^2}=1\)
\(\Leftrightarrow\left|x-2\right|=1\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2=1\\x-2=-1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\\x=1\end{matrix}\right.\)
b, ĐK: \(x\ge3\)
\(pt\Leftrightarrow\sqrt{x-3}\left(\sqrt{x}-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x-3}=0\\\sqrt{x}-1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\left(tm\right)\\x=1\left(l\right)\end{matrix}\right.\)
\(a,=12x^3y-4x^2y^2+3xy^3\\ b,=x^3+3x^2-5x+3x^2+9x-15-x^3-4x^2+4x\\ =2x^2+8x-15\)
b: Ta có: \(\left(x+3\right)\left(x^2+3x-5\right)-x\left(x-2\right)^2\)
\(=x^3+3x^2-5x+3x^2+9x-15-x^3+4x^2-4x\)
\(=10x^2-15\)
Bài 1:
b: \(=\dfrac{x+3-4-x}{x-2}=\dfrac{-1}{x-2}\)
Bài 2:
a: \(=\dfrac{x+1}{2\left(x+3\right)}+\dfrac{2x+3}{x\left(x+3\right)}\)
\(=\dfrac{x^2+x+4x+6}{2x\left(x+3\right)}=\dfrac{x^2+5x+6}{2x\left(x+3\right)}=\dfrac{x+2}{2x}\)
d: \(=\dfrac{3}{2x^2y}+\dfrac{5}{xy^2}+\dfrac{x}{y^3}\)
\(=\dfrac{3y^2+10xy+2x^3}{2x^2y^3}\)
e: \(=\dfrac{x^2+2xy+x^2-2xy-4xy}{\left(x+2y\right)\left(x-2y\right)}=\dfrac{2x^2-4xy}{\left(x+2y\right)\cdot\left(x-2y\right)}=\dfrac{2x}{x+2y}\)
a) Ta có x 6 + 2 x 3 + 3 x 3 − 1 . 3 x x + 1 . x 2 + x + 1 x 6 + 2 x 3 + 3 = 3 x x 2 − 1
b) Gợi ý: a 3 + 2 a 2 - a - 2 = (a - 1)(a + 1) (a + 2)
Thực hiện phép tính từ trái qua phải thu được: = 1 3