id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
4,068 |
\left(1 + X^{10}\right)\cdot (X^{10} + (-1)) = (-1) + X^{20}
|
33,265 |
\mathbb{E}(X_1) + \mathbb{E}(X_2) = \mathbb{E}(X_2 + X_1)
|
30,531 |
( w, x) + ( 0, 1) = ( w + 0, x) = [w, x]
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-2,185 |
\dfrac{5}{12} = -1/12 + 6/12
|
21,080 |
6 \cdot x = 3 \cdot x + x \cdot 3
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21,472 |
\mathbb{Cov}[x_1, x_2] = \mathbb{E}[x_1 \cdot x_2] - \mathbb{E}[x_1] \cdot \mathbb{E}[x_2] = \mathbb{E}[x_1 \cdot x_2]
|
27,355 |
2^{k + 1 + 1} = 2 \cdot 2^{k + 1} > 2 \cdot k \cdot k
|
7,934 |
\frac{1}{0!\cdot 0!\cdot 3!}\cdot 3! = 1
|
22,352 |
g^2 + a^2 + 2 \cdot a \cdot g = (a + g)^2
|
3,284 |
J*(d + e) = J*d + e*J
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-24,079 |
\frac{126}{6 + 8} = \frac{1}{14}126 = \tfrac{1}{14}126 = 9
|
15,442 |
C \cdot (B + A) = AC + CB
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-20,168 |
\tfrac{1}{54 \cdot \left(-1\right) + 54 \cdot r} \cdot (9 - r \cdot 9) = -\frac16 \cdot \frac{9 \cdot r + 9 \cdot (-1)}{r \cdot 9 + 9 \cdot (-1)}
|
16,955 |
\tfrac{1}{3^y}*5^y = \left(\frac53\right)^y
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26,547 |
\sin(\theta)=\frac{\sin^2(\theta)}{\sin(\theta)}=\sin(\theta)
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14,065 |
x^3 + 3x^2 + 6 = (x + 1)^3 - 3x + (-1) + 6 = (x + 1) \cdot (x + 1) \cdot (x + 1) - 3\left(x + 1\right) + 8
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8,387 |
\frac1{ue^u}=-6\Rightarrow ue^u=-\frac16
|
-4,641 |
\frac{1}{x + 4}*5 - \dfrac{3}{x + 5} = \dfrac{1}{20 + x^2 + x*9}*\left(2*x + 13\right)
|
14,183 |
\frac{2}{1 - \dfrac{1}{3}} = 3
|
9,393 |
\frac{1}{11}121\cdot \left(1 + 10\right) = 121
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21,391 |
\dfrac{2 \cdot z}{z^2 + 1} = \frac{2 \cdot z}{(z + i) \cdot (z - i)} = (z + i)^{-1} + (z - i)^{-1}
|
3,006 |
x^3 - \eta^3 = (-\eta + x) \cdot (\eta^2 + x \cdot x + x \cdot \eta)
|
21,468 |
9 + t \cdot t + 2\cdot t = 8 + (1 + t)^2
|
16,852 |
4^{n + 1} + (-1) = 4*4^n + (-1) = 3*4^n + 4^n + (-1)
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15,523 |
x^2 + y * y - x*y = -(-y*x + x^2) + x^2 + x^2 - 2*y*x + y^2
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-4,471 |
\frac{7\cdot (-1) + x}{10 + x^2 + x\cdot 7} = \frac{1}{x + 5}\cdot 4 - \frac{3}{2 + x}
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47,257 |
\cos^{12}(x) = (\cos^2(x))^6 = \sin^6\left(x\right)
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21,749 |
\left(s + (-1)\right) \cdot s = s^2 - s
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18,812 |
W_0/6 = \frac{6}{36}\cdot W_0
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19,005 |
\frac{\xi}{x} = (x + n \xi)/x = \dfrac{n}{x \cap n \xi} \xi
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-26,445 |
(5\cdot x + 2\cdot \left(-1\right))\cdot 8 = 8\cdot \left(\frac{40\cdot x}{8}\cdot 1 - \tfrac{16}{8}\right)
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18,886 |
h + e := h + e
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29,353 |
2\cdot (-1) + 13 + \left(-1\right) = 10
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14,413 |
(a^3)^n + n^3 = (a^n)^2 * a^n + n * n^2 = (a^n + n)*((a^2)^n - n*a^n + n^2)
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-4,478 |
(y + 1)*(y + 5*(-1)) = y * y - y*4 + 5*(-1)
|
-17,078 |
6 = 6 \cdot \left(-t\right) + 6 \cdot \left(-4\right) = -6 \cdot t - 24 = -6 \cdot t + 24 \cdot (-1)
|
10,680 |
\frac{3600}{5040} = \dfrac{1}{7!}(-2! \cdot 6! + \frac{1}{8}8!)
|
-23,049 |
21/18 = \frac{3\times 7}{6\times 3}
|
-1,496 |
\tfrac{18}{45} = \frac{18\cdot \dfrac{1}{9}}{45\cdot 1/9} = 2/5
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-27,237 |
\sum_{l=1}^\infty \frac{1}{l\cdot 2^l}\cdot \left(1 + 3\cdot \left(-1\right)\right)^l = \sum_{l=1}^\infty \frac{1}{l\cdot 2^l}\cdot \left(-2\right)^l = \sum_{l=1}^\infty \frac{(-1)^l\cdot 2^l}{l\cdot 2^l} = \sum_{l=1}^\infty \frac{(-1)^l}{l}
|
3 |
(d + e*10) (a*10 + g) = d g + 100 a e + (e g + d a)*10
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40,081 |
1 + \frac{1}{1 + \frac{1}{1 + 1/(3\cdot 1/2)}} = 1 + \frac{1}{1 + \frac{1}{\frac{1}{1 + 1/2} + 1}}
|
22,093 |
93.5 = 0.4 \times (0 + 95 + 0 \times \left(-1\right)) + \left(95 + 0 \times (-1) + 5\right) \times 0.1 + 0.4 \times (0 + 95 + 5 \times (-1)) + (5 + 95 + 5 \times \left(-1\right)) \times 0.1
|
17,614 |
\cos\left(z_1 + iz_2\right) = \cos{z_1} \cos{iz_2} - \sin{z_1} \sin{iz_2} = \cos{z_1} \cosh{z_2} - i\sin{z_1} \sinh{z_2}
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-26,632 |
64\cdot y^6 - y^3\cdot 48 + 9 = (8\cdot y^3 + 3\cdot \left(-1\right)) \cdot (8\cdot y^3 + 3\cdot \left(-1\right))
|
31,642 |
\frac{\partial}{\partial x} \sin\left(G + x*m\right) = \cos(x*m + G)*m
|
-20,525 |
2/9*\dfrac{1}{p + 8}*(8 + p) = \frac{p*2 + 16}{72 + p*9}
|
19,968 |
q^4 - q^3 - q^2 + q = (-q + q^2) \cdot (q \cdot q + (-1))
|
15,097 |
2^{k + 1} \cdot (k + 1) = 2^{1 + k} + 2^{1 + k} \cdot k
|
-24,890 |
\dfrac16 = \tfrac{p}{6\pi}*6\pi = p
|
5,627 |
25 \cdot 25 = 65 \cdot 65 - 60^2
|
-4,923 |
0.53\cdot 10^{(-1) (-1) + 1} = 0.53\cdot 10 \cdot 10
|
28,725 |
(x + 1)^l*(1 + x)^l = \left(1 + x\right)^{2l}
|
-20,061 |
-3/1 \frac{1}{\left(-8\right) q}(q*(-8)) = \dfrac{1}{q*\left(-8\right)}24 q
|
12,464 |
((-1)*0.5 + I)^2 = 0.25 + I^2 - I
|
-6,640 |
\dfrac{1}{2\left(10 (-1) + t\right)}4 = \tfrac{4}{2t + 20 (-1)}
|
-18,257 |
\dfrac{s^2 + 4\cdot s}{32 + s^2 + s\cdot 12} = \frac{s}{(s + 4)\cdot (s + 8)}\cdot (4 + s)
|
-24,350 |
\frac{90}{6 + 4} = \frac{90}{10} = 90/10 = 9
|
26,160 |
\frac{1}{2^n}\cdot n\cdot 2^{(-1) + n} = n/2
|
22,996 |
-2/m + \dfrac{1}{m + (-1)} + \frac{1}{m + 1} = \frac{1}{-m + m^3} \cdot 2
|
-3,800 |
\frac{1}{q^4}\cdot q\cdot \frac{7}{4} = \frac{q\cdot 7}{q^4\cdot 4}
|
31,516 |
(v^X\times z^X\times m)^X = m^X\times z\times v = v^X\times z^X\times m
|
17,208 |
9 \cdot (-16) = -144
|
-3,419 |
\sqrt{2} \cdot (2 + 1 + 3) = 6 \cdot \sqrt{2}
|
30,587 |
f + x + c = x + c + f
|
23,453 |
i^4 = i^{2 + 2} = i^2*i^2 = \left(-1\right)*(-1) = 1
|
2,588 |
\frac{x_l}{x_l + 1} = x_l - \frac{x_l^2}{1 + x_l}
|
32,390 |
n*n! + n! = (n + 1)!
|
-2,979 |
\sqrt{7} = (4 + (-1) + 2 (-1)) \sqrt{7}
|
36,788 |
0.2399999*\dotsm = 0.24
|
10,993 |
2.86 \cdot 10^{18} \cdot 0.972 \cdot 0.96^2 \cdot 0.9996 \cdot 0.95 = 20!
|
24,301 |
\int b\,dz = \int b\,dz
|
9,303 |
x^3 - 4x + 4(-1) = \left(x + 3(-1)\right) (x^2 + 3x + 5) = (x + 3(-1)) (x^2 - 8x + 5) = (x + 3\left(-1\right)) \left(x + 4(-1)\right)^2
|
-1,823 |
\dfrac{5}{4} \cdot \pi = -2 \cdot \pi + \tfrac{13}{4} \cdot \pi
|
-3,987 |
6 \cdot \dfrac{1}{a^2} = \dfrac{6}{a^2}
|
15,565 |
\frac{5 -5}{6} = 0
|
15,051 |
r \cdot r = (5 + (-1))^2 + (4 + 3(-1)) \cdot (4 + 3(-1)) + (2 + 2)^2 \Rightarrow r = 33^{1/2}
|
-20,367 |
\frac{36}{9 \cdot x} \cdot x = 4/1 \cdot \frac{x}{9 \cdot x} \cdot 9
|
34,315 |
60 = 3^1\cdot 5^1\cdot 2^2
|
9,753 |
-h \cdot 12 + 1 = 0 \implies \dfrac{1}{12} = h
|
-19,623 |
\frac{1}{5/9\cdot 6} = 9\cdot 1/5/6
|
-10,435 |
-\frac{5}{4 \cdot \eta + 2 \cdot (-1)} \cdot 5/5 = -\dfrac{1}{10 \cdot (-1) + 20 \cdot \eta} \cdot 25
|
932 |
m^2 \cdot m + \frac{m^4}{4} = \left(m^4 + 4\cdot m^3\right)/4
|
14,523 |
17 \cdot 17 - 16^2 = 289 + 256 \cdot (-1) = 33 = 49 + 16 \cdot (-1) = 7 \cdot 7 - 4^2
|
27,156 |
\sigma_t^2/2 + x_t = 0 \implies x_t = -\frac{\sigma_t^2}{2}
|
12,087 |
0 \gt -a + r \Rightarrow r \lt a
|
-7,082 |
3/28 = \frac{3 / 4}{7} \cdot 1
|
34,304 |
33 \cdot 4^2 = 528
|
8,459 |
\dfrac{1}{\sqrt{I}\cdot 2} = \frac{d}{dI} \sqrt{I}
|
15,176 |
y \cdot f' \cdot x \cdot 2 = z_x \Rightarrow x \cdot f' \cdot y \cdot y \cdot y \cdot 2 = y^2 \cdot z_x
|
-3,865 |
\frac{3}{x^3} \cdot \frac{1}{2} = \frac{3}{2 \cdot x^3}
|
-18,321 |
\frac{1}{a^2 + 2a}(14 (-1) + a^2 - a \cdot 5) = \frac{1}{a \cdot (2 + a)}(a + 2) (7(-1) + a)
|
31,650 |
\sum_{j=0}^n \binom{n}{j} = \sum_{j=0}^n \binom{n}{j}
|
22,615 |
\tfrac{1}{24}\times 6 = 1/4
|
27,237 |
\frac{\mathrm{d}}{\mathrm{d}x} \arcsin(x) = \frac{1}{(1 - x^2)^{1/2}}
|
16,042 |
-\frac1n + f/b = \frac{n \cdot f - b}{b \cdot n}
|
10,493 |
1/18 = 1/(3*9) + \dfrac{2*1/3}{36}
|
33,104 |
31/32 = 1 - \left(1/2\right)^5
|
-5,372 |
2.03*10 = 2.03*10/1000 = \frac{2.03}{100}
|
-481 |
\pi \frac{21}{2} - 10 \pi = \pi/2
|
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