id
int64
-30,985
55.9k
text
stringlengths
5
437k
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
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