id
int64 -30,985
55.9k
| text
stringlengths 5
437k
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|---|---|
8,559
|
-\frac{7!}{6!} + \dfrac{8!}{6!} = 49
|
-4,151
|
96/132 \cdot \frac{1}{y^4} \cdot y = \tfrac{y}{y^4 \cdot 132} \cdot 96
|
39,175
|
2 \cdot 2 \cdot 2 \cdot 43 = 344
|
31,026
|
1\% = \dfrac{1}{100}
|
-2,829
|
2^{1 / 2} + (9*2)^{\frac{1}{2}} = 18^{1 / 2} + 2^{1 / 2}
|
5,261
|
r_1 \times r_2 = 2/3 \Rightarrow r_2 = \frac{1}{r_1 \times 3} \times 2
|
18,686
|
z^2 \coloneqq zz
|
-1,966
|
\dfrac14\cdot \pi + \pi = \pi\cdot \frac{5}{4}
|
-30,256
|
\dfrac{1}{y + 6 \cdot \left(-1\right)} \cdot (y^2 + 36 \cdot (-1)) = \frac{1}{y + 6 \cdot \left(-1\right)} \cdot \left(y + 6\right) \cdot \left(y + 6 \cdot (-1)\right) = y + 6
|
16,377
|
(2 + \omega)^2 - \omega^2 = \omega\cdot 4 + 4
|
26,443
|
\left(l \cdot l + i^2 \cdot 7 = x^2 \Rightarrow x \cdot x - l^2 = 7i^2\right) \Rightarrow (x + l) (x - l) = i^2 \cdot 7
|
-26,356
|
1/16 = -\frac{1}{4}\times (-\frac{1}{4})
|
-20,470
|
100/\left(-70\right) = -10/(-10)*(-\frac17*10)
|
-20,010
|
\dfrac{q + 6}{3 \cdot (-1) - 2 \cdot q} \cdot \frac{1}{4} \cdot 4 = \frac{1}{-q \cdot 8 + 12 \cdot (-1)} \cdot (4 \cdot q + 24)
|
9,533
|
x + y + z = 4\Longrightarrow x + z = -y + 4
|
-24,103
|
6 + \frac18 \times 48 = 6 + 6 = 6 + 6 = 12
|
34,998
|
6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6 = 46656
|
7,649
|
\dfrac{b}{x} = \dfrac{b}{x}
|
29,574
|
\left(x^2 + z^2\right)^2 = x^4 + 2\cdot x^2\cdot z^2 + x^4 \geq x^4 + z^4
|
-19,622
|
\frac{8*\frac13}{1/3*5} = 8/3*3/5
|
3,430
|
A^2 + A \times z \times 2 + z^2 = \left(A + z\right)^2
|
-23,393
|
\frac{1}{35} \cdot 9 = \dfrac35 \cdot \frac{1}{7} \cdot 3
|
25,825
|
\mathbb{E}\left[U^2\right] - \mathbb{E}\left[U\right]^2 = \mathbb{E}\left[\left(U - \mathbb{E}\left[U\right]\right)^2\right]
|
34,155
|
n + n = 2\cdot n
|
22,322
|
\frac423 = 6
|
12,012
|
1 - -\frac{1}{2 \cdot \left(1 + 1 + (-1)\right)} - -\tfrac{1}{(1 + 1 + 1) \cdot 2} = 5/3
|
17,580
|
2\left(l + 1\right) = 2l + 2
|
-4,285
|
\frac65 = \dfrac{1}{5}6
|
5,439
|
\left(12\times (-1) + d\right)^2 = d \times d - 24\times d + 144
|
-30,872
|
54 = 9\times 6
|
6,582
|
0 = 8 \cdot c - a \cdot 16 \implies c = 2 \cdot a
|
54,474
|
5^{2012} = (5^4)^{503} = (313*2 + (-1))^{503} = (313*2)^{513}
|
17,951
|
p^2 = (p + 3\cdot (-1) + 3)^2 = (p + 3\cdot \left(-1\right))^2 + 2\cdot (p + 3\cdot (-1))\cdot 3 + 3 \cdot 3 = \left(p + 3\cdot (-1)\right)^2 + 6\cdot \left(p + 3\cdot (-1)\right) + 9
|
35,876
|
3/1 = 3 \cdot (-1) + p \Rightarrow 6 = p
|
8,327
|
\left(e = e^{-4B + 4} \Rightarrow 4 - 4B = 1\right) \Rightarrow B = 3/4
|
-4,355
|
t^3/(t t) = \frac{t}{t t} t t = t
|
8,277
|
g + x + o = g + x + o
|
14,088
|
r = x\cdot \frac1r/(z\cdot 1/r)\cdot \tfrac1r\cdot x = x^2/(z\cdot r)
|
31,748
|
x = \sin\left(\arcsin{x}\right)
|
17,294
|
5/34*5/34*34 = 25/34
|
33,605
|
k + 2\cdot (-1) + 3 = k + 1
|
16,868
|
(2 - \sqrt{x}) \cdot \left(2 + \sqrt{x}\right) = -x + 4
|
-20,709
|
-6/5 \cdot \dfrac{t \cdot 9 + 4 \cdot (-1)}{t \cdot 9 + 4 \cdot (-1)} = \frac{1}{20 \cdot (-1) + t \cdot 45} \cdot (24 - t \cdot 54)
|
13,658
|
123456789 + 111111111\cdot (-1) = 999999999 + 987654321\cdot (-1)
|
29,905
|
15 = \dfrac{1}{2} \cdot (31 + (-1))
|
17,282
|
-(p^3 \cdot 2 + p)/3 + p^3 = (-p + p^3)/3
|
31,815
|
3 + 4\cdot 15 = 63
|
2,407
|
\mathbb{E}(Y)^2 + Var(Y) = \mathbb{E}(Y^2)
|
14,222
|
1 - 3/7 = \frac{1}{7}*4
|
26,367
|
3 \cdot 3\cdot 2^8\cdot 5^7 = 10^6\cdot 6^2\cdot 5
|
-11,471
|
i \cdot 7 + 6 + 20 = 7 \cdot i + 26
|
20,503
|
(W^2)^{1/2} = |W| = -W
|
38,792
|
\var\left( \frac{X_1+\cdots+X_n}{n} \right) = \frac{1}{n^2} \var(X_1+\cdots+X_n) = \frac{1}{n^2}\left(\var(X_1)+\cdots+\var(X_n)\right)
|
-9,571
|
\dfrac44 = 1
|
21,467
|
\left(n + x\right) \cdot \left(n + x\right) = n^2 + x^2 + 2\cdot n\cdot x
|
23,286
|
\cos{k*z}*(1 + i*\tan{k*z}) = e^{i*k*z} = (e^{i*z})^k
|
25,997
|
y + (-1) = (\sqrt{y} + (-1)) (\sqrt{y} + 1)
|
22,226
|
g/x\cdot \tfrac{c}{d} = g\cdot c/(x\cdot d)
|
28,712
|
\frac{1}{48} + 1/6 = 1/8 + 1/16
|
33,771
|
0 = y^2 + 3 \times y = y \times (y + 3)
|
-2,966
|
-\sqrt{99} + \sqrt{176} = \sqrt{16\cdot 11} - \sqrt{9\cdot 11}
|
76
|
y \cdot p_1 + y \cdot p_2 = (p_1 + p_2) \cdot y
|
12,321
|
\frac12 \cdot (g_1 + 0) = g_2,\frac12 \cdot (b + 0) = j rightarrow g_1 = g_2 \cdot 2, j \cdot 2 = b
|
51,046
|
\int \tfrac{1}{y}\cdot \sqrt{1 + y^2}\,\text{d}y = \int \frac{1}{y\cdot \sqrt{1 + y^2}}\cdot \left(1 + y \cdot y\right)\,\text{d}y = \int \frac{1}{y\cdot \sqrt{1 + y \cdot y}}\,\text{d}y
|
-14,302
|
10 \times (8 + 2) = 10 \times 10 = 100
|
18,083
|
-3\cdot (1 + 2 + 3 + 4 + 5 + \dots) = \frac{1}{4}
|
6,497
|
x + 1 + (x + 2\cdot (-1))^m < 2\cdot x^2 - x = x + 2 \Rightarrow \left(x + 2\cdot (-1)\right)^m \lt 1
|
-12,134
|
\frac{11}{20} = s/\left(16*\pi\right)*16*\pi = s
|
-2,832
|
125^{1/2} + 80^{1/2} = (25\cdot 5)^{1/2} + (16\cdot 5)^{1/2}
|
6,347
|
(z + (-1)) (1 + z^2 + z) = z \cdot z \cdot z + (-1)
|
-1,719
|
-\pi\times 7/6 = -\pi\times \frac{1}{6}\times 7 + 0
|
-5,573
|
\dfrac{x*2}{x * x - x + 2*(-1)} = \frac{2*x}{(x + 1)*(x + 2*\left(-1\right))}*1
|
34,213
|
y = e y = e^2 e y = (y^5)^3 y = y^{16} = (y^2)^8
|
-2,156
|
-\frac{1}{3}*5*\pi + \dfrac{1}{6}*\pi = -\pi*\dfrac12*3
|
-7,292
|
\frac18 \cdot 4 \cdot 5/9 = \frac{5}{18}
|
11,515
|
\arccos(4/5) = y \Rightarrow \cos(y) = 4/5,\sin(y) = \sqrt{1 - (4/5)^2} = \frac{3}{5}
|
77
|
(a * a - ba*2 + 2b^2) \left(a^2 + 2ab + 2b^2\right) = a^4 + b^4*4
|
10,500
|
y \times (n + 1) = y \times n + y
|
2,513
|
Z^k \cdot y = Z \cdot y \cdot Z^{(-1) + k}
|
-22,832
|
130/39 = 10\cdot 13/\left(13\cdot 3\right)
|
9,785
|
\left(y + 2\right)\cdot (y + 3\cdot (-1))\cdot ((-1) + y) = 6 + y^3 - y^2\cdot 2 - y\cdot 5
|
28,106
|
\|\lambda*x\| = \|\lambda*x - V*x + V*x\| \leq \|\lambda*x - V*x\| + \|V*x\|
|
-20,607
|
\dfrac{-7}{5} \times \dfrac{5z - 1}{5z - 1} = \dfrac{-35z + 7}{25z - 5}
|
18,242
|
\ln(B) = \ln(A)\Longrightarrow A = B
|
-3,069
|
\sqrt{11}\cdot 4 = \sqrt{11}\cdot (5 + (-1))
|
-104
|
-23 + 5 \cdot \left(-1\right) = -28
|
32,244
|
3 (x + f) = 3 f + 3 x
|
35,820
|
{4 \choose 1} = {4 \choose 3} = 4
|
39,913
|
\frac{2\cdot 2^{1/2}}{2^{1/2} + 2} = -2 + 2^{1/2}\cdot 2
|
-1,899
|
-\dfrac12\cdot \pi + \pi\cdot 4/3 = \pi\cdot \frac16\cdot 5
|
50,675
|
1/\sin{2 x} + \frac{1}{\tan{2 x}} = \dfrac{1}{2 \sin{x} \cos{x}} (2 \cos^2{x} + (-1) + 1) = 1/\tan{x}
|
12,219
|
\tfrac{1}{x_1} \cdot (S_1 - x_1) = \frac14 \Rightarrow S_1 = \frac54 \cdot x_1
|
3,919
|
\sin(A + B) = \cos{B} \cdot \sin{A} + \cos{A} \cdot \sin{B}
|
10,989
|
x^n = x^{2 \cdot n/2} = \left(x^{2 \cdot n}\right)^{\dfrac12} = \sqrt{x^{2 \cdot n}}
|
35,459
|
R_l/(C_l) = \tfrac{R_l}{C_l}
|
10,928
|
z^2 + z + 1 = (z + 2) (z + 2) = (z + \left(-1\right)) (z + (-1))
|
23,564
|
95^8 = 95 \cdot 95 \cdot \dotsm
|
1,659
|
(-1) + \cos^2{y} \cdot 2 = \cos{y \cdot 2} \implies \frac12 \cdot (\cos{2 \cdot y} + 1) = \cos^2{y}
|
-15,400
|
\frac{\frac{1}{k^6} z^{15}}{k^8 \frac{1}{z^{20}}} = \frac{1}{k^6 k^8} \frac{z^{15}}{\frac{1}{z^{20}}} = \tfrac{1}{k^{14}}z^{15 - -20} = \frac{1}{k^{14}}z^{35}
|
26,160
|
\frac{1}{2}*x = \frac{x}{2^x}*2^{x + (-1)}
|
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