id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
206 |
x \times G + y \times x + G \times y = x \times y \times G \times (\tfrac{1}{G} + 1/x + 1/y)
|
10,294 |
(-1/2)^{1 + k} = \dfrac{\left(-1\right)^{1 + k}}{2^{k + 1}}
|
7,453 |
z^2\cdot x\cdot 3 + x^3 + z^3 + 3\cdot x \cdot x\cdot z = \left(x + z\right) \cdot \left(x + z\right) \cdot \left(x + z\right)
|
-22,970 |
\frac{6 \cdot 5}{6 \cdot 4} = \frac{30}{24}
|
6,699 |
(a + b)/2 = (2\cdot a + b - a)/2 = a + \tfrac12\cdot \left(b - a\right)
|
2,318 |
-z + x\cdot z + x = 1 + (\left(-1\right) + x)\cdot (1 + z)
|
-3,647 |
\frac{r^4}{r^2} = rr r r/(rr) = r \cdot r
|
36,593 |
4.5! = \dfrac{945}{32} \cdot \sqrt{\pi} \approx 29.53125
|
3,001 |
\frac{1}{a*c} = \frac{1}{a*c}
|
41,569 |
8 = 2\cdot (3 + 1)
|
9,662 |
\tfrac{1}{l^3} = \dfrac{1}{l^3}
|
4,173 |
-7^2 + 8^2 = -1^2 + 4^2
|
-23,654 |
5/12 = \frac{\tfrac16}{2} \cdot 5
|
18,625 |
2*(1^k + 2^k + \dots + l^k + \left(1 + l\right)^k) - 2*(1^k + 2^k + \dots + l^k) = 2*(1 + l)^k
|
-4,852 |
10^{2 - 2} \cdot 8 = 10^0 \cdot 8
|
40,934 |
\tfrac{5!}{5} = 4!
|
26,636 |
3^2 + 4 \cdot 4 + 12^2 + 84^2 + 3612^2 = 3613 \cdot 3613
|
24,325 |
\frac{1}{2^{10}} \cdot 144 = \frac{1}{2^{10}} \cdot (1 + 10 + 36 + 56 + 35 + 6)
|
16,949 |
\frac{1}{2}\cdot (3^m + (-1)) = 1 + 3 + 3^2 + \ldots + 3^{m + (-1)}
|
23,117 |
(2\cdot k + 1)^2 + 2\cdot k + 1 + 1 = 4\cdot k^2 + 6\cdot k + 2 + 1 = 2\cdot (2\cdot k \cdot k + 3\cdot k + 1) + 1
|
12,236 |
3 + n\cdot 6 = (2\cdot n + 1)\cdot 3
|
25,629 |
\mathbb{E}[Q + Y] = \mathbb{E}[Q] + \mathbb{E}[Y]
|
1,880 |
\int \sqrt{2^2 - y \cdot y}\cdot y^3\,\text{d}y = \int y^3\cdot \sqrt{(2 - y)\cdot (y + 2)}\,\text{d}y
|
-6,708 |
80/100 + \frac{4}{100} = \frac{1}{100}\cdot 4 + \dfrac{8}{10}
|
12,298 |
1 = (y \cdot b)^3 = y^3 \cdot b^{y^2} \cdot b^y \cdot b = b^{\frac1y} \cdot b^y \cdot b
|
-3,558 |
1/(2\cdot t) = \frac{1}{2\cdot t}
|
33,630 |
t^9 = (t^3)^3 = (t + 1) \cdot (t + 1)^2 = t^3 + 1 = t + 2 = t + (-1)
|
-9,441 |
x\cdot 40 = 2\cdot 2\cdot 2\cdot 5\cdot x
|
-26,194 |
1 = 9 + 8(-1)
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1,453 |
1638 = 2\times 3^2\times 7\times 13 = (1^2 + 1^2)\times 3^2\times (2^2 + 1^2 + 1^2 + 1^2)\times (3 \times 3 + 2^2)
|
2,446 |
\dfrac{1}{36}5 = \frac{4}{8}*5/10*\dfrac{1}{9}5
|
20,831 |
x*\dfrac{b}{x}/x = b/x = b/x
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6,662 |
-8 = 8 \cdot (\cos(\pi) + i \cdot \sin(\pi))
|
4,924 |
\frac{h^{\dfrac12}}{h} = \frac{h^{\frac12}}{h} = h^{-1/2}
|
-22,222 |
z \cdot z + 9 \cdot z + 20 = (5 + z) \cdot (z + 4)
|
4,444 |
\left( x, z\right) \cap ( y, z) = \left( x*y, y*z, x*z, z\right) = ( x*y, y*z, z)
|
-1,714 |
\frac165 \pi = -\pi \frac{5}{12} + \frac{5}{4} \pi
|
13,242 |
36 + 36 \cdot k \cdot k + 36 \cdot k = 27 + \left(k \cdot 6 + 3\right)^2
|
22,105 |
\mathbf{N} := \left\{\ldots, 2, 3, 1\right\}
|
6,918 |
x + (z + 2 \cdot \left(-1\right))^2 = 0 \Rightarrow z = 2 \pm (-x)^{1/2}
|
43,484 |
2 + 0.9 + 0.99 + 0.999 + ... = 3 \lt π
|
-1,720 |
\pi \cdot 23/12 + \pi \cdot \frac{1}{12} \cdot 19 = \frac{7}{2} \cdot \pi
|
-20,263 |
\frac{1}{72 \cdot (-1) - f \cdot 27} \cdot (64 \cdot \left(-1\right) - f \cdot 24) = 8/9 \cdot \frac{1}{-3 \cdot f + 8 \cdot (-1)} \cdot (-3 \cdot f + 8 \cdot \left(-1\right))
|
10,328 |
\cos^3{y} - 3 \cdot \sin^2{y} \cdot \cos{y} = \cos{3 \cdot y}
|
-18,617 |
5\cdot t + 5 = 7\cdot \left(3\cdot t + 9\right) = 21\cdot t + 63
|
7,425 |
a \cdot x \cdot y + b \cdot y \cdot z = (a \cdot x + b \cdot z) \cdot y \leq \sqrt{a \cdot a + b^2} \cdot \sqrt{x^2 + z^2} \cdot |y|
|
31,645 |
66 + \left(-1\right) + 6\cdot \left(-1\right) + 6\cdot \left(-1\right) + 15\cdot (-1) = 38
|
-15,928 |
-\frac{83}{10} = -9/10\cdot 10 + \frac{7}{10}
|
22,916 |
\dfrac{39!}{5! \cdot (5 \cdot (-1) + 39)!} = 575757
|
12,857 |
\mathbb{E}(TB) = \mathbb{E}(T) \mathbb{E}(B)
|
4,365 |
\tfrac{6}{7} = 6/7
|
-16,567 |
2\cdot 25^{\frac{1}{2}}\cdot 11^{1 / 2} = 2\cdot 5\cdot 11^{\dfrac{1}{2}} = 10\cdot 11^{\dfrac{1}{2}}
|
-25,127 |
\frac{x\cdot 36 + 75\cdot x \cdot x}{\sqrt{3 + 5\cdot x}\cdot 2} = \frac{d}{dx} (3\cdot x^2\cdot \sqrt{x\cdot 5 + 3})
|
-18,954 |
3/4 = \frac{H_q}{4*\pi}*4*\pi = H_q
|
22,003 |
30/31/30 + \frac{1}{31} = \frac{1}{31}\cdot 2
|
-20,858 |
-\frac{5}{-5} (-\frac89) = \frac{1}{-45}40
|
-17,991 |
30 - 23 = 7
|
11,428 |
f\cdot h_n\cdot c = h_n\cdot c\cdot f
|
36,054 |
\dfrac{\sqrt{30}}{5} \cdot 2 = \frac25 \cdot \sqrt{30}
|
21,795 |
\dfrac{4}{2\cdot (2\cdot t + 1)} = \frac{2}{2\cdot t + 1} = \tfrac{1}{t + 1} + \frac{1}{(t + 1)\cdot (2\cdot t + 1)}
|
6,001 |
(s + 37) \cdot (37 \cdot (-1) + s) = -37^2 + s^2
|
46,837 |
\left(2*r + (-1)\right)^2 - r * r - 1/3 = 3*r^2 - 4*r + 4/3 = 3*(r - 2/3)^2
|
-559 |
(e^{7 \cdot i \cdot \pi/12})^{17} = e^{7 \cdot \pi \cdot i/12 \cdot 17}
|
10,671 |
1/15 = \frac{12}{180}
|
4,664 |
-4\cdot b\cdot f + (f + b) \cdot (f + b) = (f - b)^2
|
-4,730 |
\dfrac{-y\cdot 4 + 11\cdot (-1)}{y \cdot y + y + 2\cdot (-1)} = \frac{1}{2 + y} - \frac{5}{y + (-1)}
|
37,454 |
\dfrac{1}{6} + 2/3 = 5/6
|
-20,535 |
\frac{1}{y + 9}\cdot (y + 9)\cdot (-\dfrac{9}{10}) = \dfrac{1}{y\cdot 10 + 90}\cdot \left(81\cdot (-1) - 9\cdot y\right)
|
15,627 |
[d,\omega] = 1 \Rightarrow ( d\cdot \omega, d^2) = d
|
-16,600 |
5 \cdot 8^{1 / 2} = (4 \cdot 2)^{1 / 2} \cdot 5
|
15,289 |
\left(\left(-1\right) + x\right)! = -((-1) + x)*(x + (-1))! + x!
|
44,460 |
2*z + 5 + 3 = 2*z + 8 = 46 + 3*z
|
2,507 |
\tan\left(\left((-1) \pi\right)/4\right) = \tan(\frac{\pi*3}{4})
|
13,299 |
g^3 - h * h * h = (g - h)*(g^2 + g*h + h^2)
|
-4,276 |
\frac{132 \cdot l^5}{110 \cdot l^5} = 132/110 \cdot \dfrac{1}{l^5} \cdot l^5
|
3,668 |
\tfrac{L^2*\frac15*4}{8} = L^2/10
|
26,005 |
r u v + v s u = v u \cdot (s + r)
|
10,251 |
g = -5 + g + 5
|
21,060 |
a \in H \implies H = aH
|
1,433 |
1/(9*10)*6/8*5/7*24*3 = \frac17 3
|
10,167 |
(1 + \cos\left(\theta*2\right))/2 = \cos^2(\theta)
|
24,079 |
(1 + T)\cdot ((-1) + T) = T \cdot T + (-1)
|
-11,514 |
-15 + 3\cdot i = 3\cdot i - 6 + 9\cdot (-1)
|
236 |
ab = 1 \Rightarrow ba=1
|
1,274 |
\dfrac{1}{D} = \frac1D := \frac{1}{D}
|
1,009 |
\cos{A} \sin{b} + \sin{A} \cos{b} = \sin(A + b)
|
14,614 |
\frac83 - \frac{1}{9}*16 = 8/9
|
-6,020 |
\frac{1}{y\cdot 2 + 16\cdot \left(-1\right)}\cdot 2 = \dfrac{1}{(y + 8\cdot (-1))\cdot 2}\cdot 2
|
14,612 |
1 - -\dfrac{1}{z - b_n} \cdot (z - a_n) + 1 = \frac{-a_n + z}{-b_n + z}
|
19,852 |
\dfrac1x = \bar{x}/(x\bar{x})
|
18,706 |
y \cdot y = 0\Longrightarrow y = 0
|
11,397 |
\dfrac{1}{2.25 - 1.5}*(1 - -1) = \frac{1}{0.75}*2 = \frac{8}{3} \approx 2.7
|
36,542 |
63 = 100 + 25*\left(-1\right) + 16*(-1) + 4
|
-9,137 |
-s \cdot 63 - 42 \cdot s^2 = -3 \cdot 3 \cdot 7 \cdot s - s \cdot 2 \cdot 3 \cdot 7 \cdot s
|
10,485 |
\frac{\partial}{\partial z} (d\cos{bz}) = -d\sin{bz} b
|
18,826 |
\tan(p) = \sin(p)/\cos(p)
|
10,222 |
2\cdot x_1 - x_2 - z_2 + y_2 = 0 \Rightarrow x_1\cdot 2 - x_2 - z_2 = -y_2
|
18,383 |
(x + 8\cdot (-1)) \cdot (x + 8\cdot (-1)) = x^2 - 16\cdot x + 64
|
7,580 |
(x + (-1))\cdot \left(x^{(-1) + p} + x^{p + 2\cdot \left(-1\right)} + \cdots + 1\right) = x^p + (-1)
|
1,328 |
y^2*2*y*y^2*2 = 2*2*y^{2 + 2 + 1}
|
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