id
int64 -30,985
55.9k
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stringlengths 5
437k
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24,714 |
e^b*e^a = e^{a + b}
|
1,240 |
\left(1 + n\right)\cdot (1 + n \cdot n - n) = 1 + n^3
|
-16,462 |
\sqrt{20} \cdot 7 = 7\sqrt{4 \cdot 5}
|
2,946 |
-1/\left(1/2\right) = -2
|
12,611 |
{n \choose r}^2 = {n \choose -r + n} {n \choose r}
|
-19,193 |
29/40 = \frac{1}{25 \cdot \pi} \cdot D_s \cdot 25 \cdot \pi = D_s
|
12,397 |
\frac1x \cdot (j + x) = \frac1x \cdot j + x/x = \frac{j}{x} + 1
|
8,627 |
((-1) + x) \left(x \cdot x + x + 4\right) = x^2 \cdot x + 3x + 4(-1)
|
14,320 |
\lambda^2 = 1 \implies 1 = \lambda
|
2,621 |
10^3 - 5^3 = \left(10 + 5\cdot (-1)\right)\cdot (2 \cdot 2\cdot 5 + 2\cdot 5 + 5)\cdot (10 + 5\cdot (-1))
|
29,900 |
4*15^2 = 28 * 28 + 4^2 + 10^2
|
-18,652 |
-\frac{90}{20} = - \frac{9}{2}
|
27,923 |
4 + x^4 = (x^2 + 2 - 2 \cdot x) \cdot (x \cdot 2 + x^2 + 2)
|
16,851 |
8 * 8 - 7*3 * 3 = 1
|
34,534 |
-n^2 + y^2 = 1 \Rightarrow (-n + y)\cdot (n + y) = 1
|
15,083 |
(y^2 - f^2)*(y^2 + f^2) = y^4 - f^4
|
30,503 |
x^2 + 4 \cdot (-1) = (x + 2) \cdot (x + 2 \cdot (-1))
|
14,981 |
-\frac89 + 8 = \frac{64}{9}
|
29,579 |
1 + 4*\left(-1\right) + 9 = 3 + 1 + 2
|
23,859 |
X = X + 0
|
19,722 |
(l + 1)! \cdot l = (l + 1 + 1)! + 1 = (l + 2)! + (-1)
|
-20,923 |
\tfrac{1}{-18} \cdot (16 - k \cdot 4) = \frac{1}{-9} \cdot \left(-2 \cdot k + 8\right) \cdot \frac22
|
41,092 |
7920 = 22 \cdot 360
|
-7,737 |
\frac{1}{4\cdot i + 2}\cdot (-2\cdot i + 4)\cdot \tfrac{-4\cdot i + 2}{2 - 4\cdot i} = \frac{1}{2 + 4\cdot i}\cdot (4 - 2\cdot i)
|
-22,955 |
\dfrac{18}{24} = \dfrac{3 \cdot 6}{ 4\cdot 6}
|
17,640 |
x x = \left(100 c x\right)^2 = 10000 c x^2
|
9,853 |
d^{l\cdot s}\cdot h = d^{s\cdot l}\cdot h
|
25,822 |
25.0 = 2C \Rightarrow 12.5 = C
|
19,508 |
a^2 - b^2 = a^2 + i^2*b^2 = a^2 + (i*b) * (i*b) = (a + b)*(a + i^2*b) = (a + b)*(a - b)
|
-6,561 |
\frac{2}{2*(2*(-1) + z)} = \frac{2}{2*z + 4*(-1)}
|
18,577 |
3 + 4 < 5 + 6 \Rightarrow 3\cdot 4 < 5\cdot 6
|
-20,258 |
\frac{9}{9} (q + 4)/7 = (36 + 9q)/63
|
22,971 |
\mathbb{E}\left(B\right) + \mathbb{E}\left(X\right) = \mathbb{E}\left(X + B\right)
|
7,323 |
-\frac{1}{3} + (1 - i)/4 + \frac14(i + 1) + \dfrac16 - 1/3 = 0
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35,554 |
(3 + 4 + 5 + 8 + 15 + 16 + 20 + 36)/8 = \frac{1}{8}\cdot 107 = 13.375
|
18,637 |
2/3 f + d \cdot 2 \Rightarrow f = -3d
|
-24,487 |
2\times \left(6 + 10\right) = 2\times 16 = 32
|
14,833 |
\frac{(-1) + x^k}{x + (-1)} = 1 + x + x x + \ldots + x^{k + (-1)}
|
20,630 |
\sqrt{41}*1311360 + 8396801 = (320*\sqrt{41} + 2049)^2
|
22,177 |
\tfrac{1}{12} + \frac12 + \tfrac{1}{4} + \frac16 = 1
|
1,709 |
z^2 + z + 1 = z * z - 2*z + 1 = (z + (-1))^2 = (z + 2)^2
|
27,462 |
(-1) + 14 + 6\left(-1\right) = 7
|
-654 |
33/2 \cdot \pi - 16 \cdot \pi = \frac{\pi}{2}
|
53,278 |
Var\left[f + f\right] = Var\left[2 \cdot f\right] = 4 \cdot Var\left[f\right] \geq Var\left[f\right]
|
8,971 |
{m \choose 1} + 2 \cdot {m \choose 2} = m^2
|
18,648 |
-1/128 = \frac{(-1) \cdot 1/6}{64 \cdot \frac{1}{3}}
|
15,751 |
\cos\left(2*x\right) = \cos^2\left(x\right)*2 + \left(-1\right)
|
27,828 |
t + t\cdot z^2 = z + z\cdot t^2 \Rightarrow t = z
|
22,660 |
\dfrac{1}{4.1} = 0.2439 \cdot \cdots
|
18,012 |
\frac{1}{\delta}\cdot z^m\cdot \delta = (z/\delta\cdot \delta)^m
|
20,728 |
(b - c)^2 \geq 4 \cdot (1 + b + c)^2 - 4 \cdot (b + c)^2 = 4 \cdot \left(1 + 2 \cdot b + 2 \cdot c\right) \gt 8 \cdot (b + c)
|
-17,123 |
-5 = -5*\left(-4*y\right) - 40 = 20*y - 40 = 20*y + 40*(-1)
|
46,568 |
9 = 3\left(-1\right) + 12
|
8,377 |
(\left(-1\right) + x)*(x + 0*(-1)) = -x + x * x
|
7,115 |
\frac{\partial}{\partial x} (\dfrac{w}{x}) = \frac{1}{x^2} (\frac{\mathrm{d}w}{\mathrm{d}x} x - w \frac{\mathrm{d}x}{\mathrm{d}x})
|
10,256 |
\pi = 6 \cdot \tan^{-1}{3^{\dfrac{1}{2}}/3}
|
11,531 |
\sin(\theta \cdot 2)/2 = \cos(\theta) \cdot \sin\left(\theta\right)
|
25,474 |
u*c*(S + T) = c*(T + S)*u
|
7,922 |
x*1/h/(x*1/h) = 1 = \frac{h x/h}{x}
|
32,672 |
d^h*d^f = d^{f + h}
|
50,555 |
0 * 0 = 0 \lt 2
|
41,422 |
|3| \cdot 5 \cdot 7 = 105
|
41,747 |
a_k\cdot 2 - 2\cdot b_k = 2\cdot a_k - b_k - b_k
|
3,835 |
g^2 + (-1) = (1 + g)*\left(g + (-1)\right)
|
1,399 |
1 - \frac{W \cdot x}{R \cdot W \cdot 1.1} = \frac{1}{1.1 \cdot R \cdot W} \cdot \left(1.1 \cdot R \cdot W - x \cdot W\right)
|
23,645 |
n + 1 = (1 + n)!/n!
|
10,860 |
(-a + x)*\left(x + a\right) = -a * a + x^2
|
24,984 |
\sin(z \cdot 2) = \cos(z) \sin(z) \cdot 2
|
1,176 |
a + g + e = g + e + a
|
41,784 |
10!^2 \cdot \frac{4^{10}}{31^{1/2}} < 1.37 \cdot 1.05 \cdot 0.2 \cdot 10^{19} = 0.2877 \cdot 10^{19} \lt 2.9 \cdot 10^{18}
|
22,241 |
hx = x = xh
|
12,965 |
(1 - \cos(x \cdot 2))/2 = \sin^2\left(x\right)
|
-3,753 |
\frac{64*y^2}{y^5*88} = \dfrac{y^2}{y^5}*64/88
|
12,171 |
4^n + n^4 = (2^n)^2 + \left(n^2\right)^2 = (2^n + n^2)^2 - 2\cdot 2^n\cdot n \cdot n
|
16,761 |
\frac{\partial}{\partial y} (-q + y) = -q + \frac{\mathrm{d}y}{\mathrm{d}y}
|
7,919 |
16 = -6\cdot -x\cdot x + 10\cdot x^2 \Rightarrow x^2 = 1
|
-10,008 |
0.01 \cdot (-88) = -\dfrac{1}{100} \cdot 88 = -\dfrac{1}{25} \cdot 22
|
20,984 |
\frac{a \cdot b}{a + b} = a - \frac{a^2}{a + b} = b - \dfrac{b^2}{a + b}
|
-20,874 |
\frac{t + 6}{t + 6}\cdot (-6/5) = \dfrac{1}{30 + 5\cdot t}\cdot \left(36\cdot (-1) - t\cdot 6\right)
|
7,653 |
120 = \dfrac{10!}{3!*(10 + 3*(-1))!}
|
39,780 |
\frac{99999999999999999}{100000000000000000} = 1 - 1/100000000000000000
|
-622 |
e^{\frac{5}{12} \cdot i \cdot \pi \cdot 13} = (e^{\frac{5}{12} \cdot \pi \cdot i})^{13}
|
6,288 |
\cos\left(z\right) = \cos(z/2\cdot 2)
|
-1,295 |
\dfrac{30}{15} = \frac{30 \cdot \frac{1}{15}}{15 \cdot \frac{1}{15}} = 2
|
-5,615 |
\frac{5}{(8 \cdot (-1) + q) \cdot 5} = \dfrac{1}{40 \cdot (-1) + q \cdot 5} \cdot 5
|
545 |
-(\frac{1}{3}\cdot \pi \cdot \pi + 3\cdot (-1)) + 1 = 4 - \tfrac{\pi^2}{3}
|
-15,253 |
\frac{1}{q^{10}\cdot p \cdot p \cdot p\cdot q^4} = \frac{1}{p^2 \cdot p\cdot q^4\cdot q^{10}}
|
16,945 |
x^3 = u\Longrightarrow x^2\cdot 3 = \frac{\mathrm{d}u}{\mathrm{d}x}
|
37,757 |
(4 + 1/2)\cdot \pi = \pi\cdot \frac{9}{2}
|
13,883 |
-x^6 - x^{180} + x^{90} + x^{48} = -x^6 - \left(x^6\right)^{30} + (x^6)^{15} + (x^6)^8
|
-3,340 |
\sqrt{3} = \sqrt{3}\times \left(3 + 2 + 4\times (-1)\right)
|
7,734 |
11 = (-2*3^{\frac{1}{2}} + 1) (-3^{\dfrac{1}{2}}*2 - 1)
|
32,130 |
p^{n + 1} = pp^n
|
3,574 |
\left|{B\times F + I}\right| = \left|{B\times F + I}\right|
|
35,186 |
\lambda + m\cdot \lambda^2 + m^2\cdot \lambda^3 + \ldots = \frac1m\cdot (m\cdot \lambda + \left(m\cdot \lambda\right)^2 + (m\cdot \lambda)^3 + \ldots) = \frac{1}{1 - m\cdot \lambda}
|
37,458 |
\binom{6}{5} \binom{6}{1} = 6*6 = 36
|
7,742 |
0 = y^H\cdot Z\cdot Z^H\cdot y = (Z^H\cdot y)^H\cdot Z^H\cdot y
|
9,342 |
-17\times \sqrt{5} + 38 = \left(2 - \sqrt{5}\right)^3
|
19,816 |
x^2 - 6\cdot x + 8 = (x + 2\cdot (-1))\cdot (x + 4\cdot (-1))
|
8,365 |
c\cdot (X - X\cdot c) = X\cdot c - c\cdot c\cdot X
|
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