id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
16,181
|
-\mathbb{E}[V_2] + \mathbb{E}[V_1] = \mathbb{E}[-V_2 + V_1]
|
34,214
|
\left(2j + 2\right)! = (2j + 2) \left(2j + 1\right)! = (2j + 2) \left(2j + 1\right) (2j)!
|
36,082
|
(\frac{3}{4}) * (\frac{3}{4}) = 9/16
|
-5,784
|
\frac{5}{25 + z\cdot 5} = \frac{5}{5\cdot \left(z + 5\right)}
|
10,665
|
\sin(z + \dfrac{\pi}{2}) = \cos\left(z\right)
|
5,866
|
(3 + y) \left(y + 1\right) = y^2 + 4y + 3
|
-15,264
|
\frac{n\cdot p^2}{p^8\cdot \dfrac{1}{n^8}} = \frac{n\cdot p^2}{\tfrac{1}{\frac{1}{p^8}\cdot n^8}}
|
20,858
|
\frac{\partial}{\partial B} (BT) = B\frac{\text{d}T}{\text{d}B} + T
|
20,436
|
\nu^s = \nu^s
|
5,874
|
y^2 + 4 = y^2 - (-1)*4 = y^2 - i^2*2^2 = y^2 - (2i)^2 = (y - 2i) (y + 2i)
|
19,536
|
967680 = 2^5 {10 \choose 5} \cdot 5!
|
11,201
|
\sin^2{z} = (1 - \cos{z*2})/2
|
15,470
|
\frac{1}{100*100}*2500 = \frac{1}{4}
|
36,696
|
30000 + 17496*(-1) = 12504
|
12,339
|
l = \left\{1, \dots, l\right\}
|
18,633
|
\tfrac32 = 1/2 + 1
|
-16,459
|
\sqrt{75}*8 = \sqrt{25*3}*8
|
3,463
|
\dfrac12 + \frac{y}{3} = \frac16\cdot \left(2\cdot y + 3\right)
|
526
|
\left(1 - q\right)^7 - 7*(1 - q)^6*q = (1 - q)^6*(1 - q - 7*q) = (1 - q)^6*(1 - 8*q)
|
21,189
|
1=\frac{8}{35}+\frac{12}{35}+\frac{1}{7}+\frac{2}{7}
|
13,994
|
(1 + x)^2 \left(1 + x\right) = 1 + 3 x + x^2 \cdot 3 + x^3
|
3,199
|
y_2\cdot y\cdot y_1 = y_2\cdot y_1 = y_1 = y_2\cdot y\cdot y_1
|
35,059
|
i^4 = \left(2l + 1\right)^4 = (4l^2 + 4l + 1) (4l^2 + 4l + 1)
|
-5,296
|
10^{10}*17 = 10^{6 + 4}*17
|
21,896
|
f \cdot f \cdot f - x^3 = (-x + f) \cdot (x \cdot x + f^2 + f \cdot x)
|
30,124
|
\tfrac{4}{2} \cdot 1 = \frac{1}{2} \cdot 4
|
-1,967
|
\frac{1}{4}*5*\pi = \pi/4 + \pi
|
7,241
|
v*(\beta + \alpha) = \beta*v + \alpha*v
|
12,364
|
4 \cdot 4 + 2^2 + 3 \cdot 3 = 29
|
27,219
|
\frac82*\left(6*\left(-1\right) + n\right) = 4*n + 24*\left(-1\right)
|
-5,783
|
\tfrac{1}{(x + 8*(-1))*(5 + x)}*2 = \frac{2}{x^2 - x*3 + 40*\left(-1\right)}
|
11,949
|
38*2/7 = 76/7
|
-11,548
|
11 + i \cdot 16 = -4 + 15 + i \cdot 16
|
20,950
|
b\cdot h + y \cdot y + (b + h)\cdot y = (y + h)\cdot (b + y)
|
21,230
|
\left(3 \cdot x + z + 5\right) \cdot \left(3 \cdot (-1) + x \cdot 2 + z\right) = x^2 \cdot 6 + z \cdot x \cdot 5 + z^2 + x + z \cdot 2 + 15 \cdot (-1)
|
26,861
|
0 = (-z + 4)^2 + z^2 - 2\cdot (-z + 4) - z\cdot 2 + 2\cdot (-1) \Rightarrow z^2 - 4\cdot z + 3 = \left(z + (-1)\right)\cdot (z + 3\cdot (-1)) = 0
|
47,735
|
{3\choose 2}=3
|
17,768
|
g + g \cdot \frac{3}{2} = g \cdot 5/2
|
-11,788
|
9^{-1/2} = (1/9)^{\frac12}
|
2,255
|
(-1) \cdot (-1) - -\theta + \theta + 3\left(-1\right) = 1 + \theta + \theta + 3(-1) = -2 + 2\theta = 0 \implies \theta = 1
|
15,592
|
s \cdot r \cdot r = s \cdot r \cdot r = r^3 \cdot s \cdot r = r^2 \cdot r \cdot s \cdot r = r^3 \cdot r^3 \cdot s = r^6 \cdot s = r^2 \cdot s
|
9,262
|
0 = n * n + n \Rightarrow n = -1, 0
|
17,655
|
45 = 3 \cdot (2^4 + (-1))
|
-12,679
|
6 = \dfrac{1}{2.5}*15
|
-6,688
|
60/100 + 9/100 = 9/100 + 6/10
|
-9,269
|
4\cdot y^2 + 4\cdot y^3 = y\cdot y\cdot y\cdot 2\cdot 2 + y\cdot y\cdot 2\cdot 2
|
-22,281
|
y^2 + y\cdot 2 + 35\cdot (-1) = (y + 5\cdot (-1))\cdot \left(y + 7\right)
|
1,634
|
\mathbb{E}\left(X\right)*\mathbb{E}\left(B\right) = \mathbb{E}\left(X*B\right)
|
17,602
|
U/2 = U - \dfrac{U}{2}
|
-20,964
|
\frac{1}{1}*9*\frac{1}{n + 7}*(n + 7) = \frac{63 + 9*n}{n + 7}
|
25,871
|
b b = p^2 - (p - f) (p - f) = 2 p f - f^2 \implies 2 f p = f f + b^2
|
-20,862
|
\dfrac{1}{36\cdot (-1) + 18\cdot y}\cdot \left(6 - 3\cdot y\right) = \dfrac{1}{y\cdot 3 + 6\cdot \left(-1\right)}\cdot (6\cdot (-1) + 3\cdot y)\cdot (-\frac{1}{6})
|
22,788
|
\cos{2\cdot D} - \cos{2\cdot Z} = -2\cdot \left(\sin^2{D} - \sin^2{Z}\right) = -2\cdot (\sin{D} - \sin{Z})\cdot (\sin{D} + \sin{Z})
|
2,220
|
\left(6 + 1\right)\cdot \left(1 + 12\right) = 91
|
-12,140
|
4/9 = \frac{s}{6\cdot \pi}\cdot 6\cdot \pi = s
|
23,529
|
\left(2 + (-1)\right) \cdot 2^{(-1) + k} = y \cdot 24 rightarrow \frac{1}{3} \cdot 2^{3 \cdot \left(-1\right) + k + (-1)} = y
|
3,653
|
6 \cdot x^2 + x = (6 \cdot x + 1) \cdot x
|
18,813
|
z - \sin\left(z\right) = z - z - z * z * z/6 + \cdots = \frac{z^3}{6} + \cdots
|
-24,098
|
\frac{140}{9 + 5} = \frac{140}{14} = 140/14 = 10
|
-10,568
|
\frac{10}{2 + x} \frac{1}{20}20 = \frac{1}{40 + x*20}200
|
-4,671
|
-\frac{1}{x + 2\cdot \left(-1\right)}\cdot 3 + \dfrac{1}{2 + x}\cdot 4 = \frac{x + 14\cdot \left(-1\right)}{x^2 + 4\cdot (-1)}
|
-1,581
|
23/12 \cdot \pi + \pi/6 = \frac{25}{12} \cdot \pi
|
46,704
|
20 = 2 \times 2\times 5
|
-2,575
|
5^{\frac{1}{2}} = 5^{\frac{1}{2}} \cdot \left(3 + 2 \cdot (-1)\right)
|
17,601
|
x^{p/q} = \left(x^p\right)^{\frac{1}{q}} = (x^p)^{\frac1q}
|
-19,735
|
\frac{2}{2} = \dfrac{2}{2}\cdot 1
|
13,592
|
(a + b) (a + b) = a^2 + b^2 = a + b
|
4,196
|
\left(x + g\right) (x + b) = bg + x^2 + x\cdot (b + g)
|
5,195
|
\frac{1}{M*N} = \frac{1}{N*M} = N^x*M^x = (M*N)^x
|
7,900
|
-f + y \geq 0 \Rightarrow f \leq y
|
18,121
|
y = 2 \cdot (-1) + 3 \cdot y + 2 - 2 \cdot y
|
14,352
|
2300 = y*2 \Rightarrow 1150 = y
|
27,924
|
61^2 \cdot 61\cdot 11^3\cdot 3^3\cdot 2^2 = 2013^3\cdot 2^2
|
24,828
|
\frac{q_2\cdot q_1}{q_2 + q_1} = \frac{1}{1/(q_1) + \frac{1}{q_2}}
|
809
|
z! = 10! \cdot 11 \cdot 12 \cdot \ldots \cdot z \gt 10! \cdot 11^{z + 10 \cdot (-1)}
|
-535
|
e^{18 \pi i \cdot 5/4} = \left(e^{5i\pi/4}\right)^{18}
|
-19,192
|
7/24 = A_s/\left(64*\pi\right)*64*\pi = A_s
|
17,477
|
8 \cdot (-1) + x \cdot 2 = (4 \cdot (-1) + x) \cdot 2
|
24,430
|
2^x = 2^x \cdot l + 1 \gt 2^x \cdot l
|
31,949
|
\pi \cdot 5/2 = \frac{\pi}{2} + \pi \cdot 2
|
36,189
|
\left(1 + d\right)^1 = 1 + d \geq 1 + d
|
25,059
|
b - \tfrac{1}{1 - a}\times b = -a\times \dfrac{b}{1 - a}
|
-26,580
|
81-4x^2=(9+2x)(9-2x)
|
18,688
|
5^{\tfrac{1}{2}} = 2\cdot u/x + \left(-1\right) = (2\cdot u - x)/x
|
6,468
|
\frac{d}{dt} e^E = E*e^E = e^E*E
|
12,684
|
F * F = FF
|
-20,138
|
\frac{3*(-1) + 9*x}{9*x + 3*(-1)}*\left(-\frac{4}{9}\right) = \frac{1}{81*x + 27*\left(-1\right)}*(-x*36 + 12)
|
-7,616
|
\dfrac{1}{13}\cdot (-69 + 6\cdot i + 46\cdot i + 4) = \frac{1}{13}\cdot (-65 + 52\cdot i) = -5 + 4\cdot i
|
-20,549
|
9/8 \cdot \frac{9 \cdot y}{9 \cdot y} = \frac{81}{y \cdot 72} \cdot y
|
8,237
|
f^{\frac1l x} = (f^x)^{1/l} = \left(f^{\frac1l}\right)^x
|
2,633
|
0 \cdot z = (1 + z) \cdot 0
|
20,050
|
\cos{3\cdot (\theta + \pi/3)} = \cos(2\cdot \pi + 3\cdot \theta) = \cos{3\cdot \theta}
|
6,473
|
(\frac{1}{1 + n}\cdot (1 + x) + \tfrac{x}{1 + n})/2 = \frac{2\cdot x + 1}{2 + n\cdot 2}
|
6,329
|
(z^4 + 1 + z^2) \cdot (-z^2 + 1) = 1 - z^6
|
5,061
|
x = \iint g\,ds\,dx = \iint g \cdot 1 \cdot 1\,ds\,dx
|
-13,263
|
5 + \frac12\cdot 14 = 5 + 7 = 12
|
-18,965
|
\dfrac{1}{15}*13 = G_x/(36*\pi)*36*\pi = G_x
|
-20,920
|
7/7 \frac{1}{r + 2 (-1)} (r\cdot (-1)) = \frac{1}{7 r + 14 (-1)} ((-7) r)
|
-4,847
|
\frac{7.8}{10} = \frac{10^{-1}}{100} 7.8 = \frac{1}{1000} 7.8
|
-2,014
|
-\frac{\pi}{6} + \pi\cdot 13/12 = \frac{1}{12}\cdot 11\cdot \pi
|
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