id
int64 -30,985
55.9k
| text
stringlengths 5
437k
|
|---|---|
-10,594
|
-\frac{80 + t\cdot 100}{t\cdot 60} = -\frac{1}{t\cdot 3}\cdot (t\cdot 5 + 4)\cdot \frac{20}{20}
|
27,551
|
x_1\cdot \overline{r_1} + \cdots + \overline{r_n}\cdot x_n = x_1\cdot r_1 + \cdots + x_n\cdot r_n
|
-26,809
|
\sum_{l=1}^\infty \tfrac{(l + 5) \cdot (-5)^l}{l^2 \cdot 5^l} = \sum_{l=1}^\infty \frac{(-1)^l \cdot 5^l}{l^2 \cdot 5^l} \cdot (l + 5) = \sum_{l=1}^\infty (-1)^l \cdot \frac{1}{l^2} \cdot (l + 5)
|
10,180
|
(-1) + z^2 = \left(1 + z\right) \cdot (z + (-1))
|
24,418
|
\tan{z}/\sec{z} = \frac{\sin{z}}{\frac{1}{\cos{z}}} \cdot 1/\cos{z} = \sin{z}
|
15,644
|
-y^2 + y\cdot 8 + 7(-1) = y + 3 - 10 + y^2 - 7y
|
-14,511
|
\frac{12}{6 + 3*\left(-1\right)} = \frac{12}{3} = \dfrac{12}{3} = 4
|
-23,406
|
3 \cdot \frac{1}{5}/5 = 3/25
|
-714
|
\left(e^{i \cdot \pi/6}\right)^{18} = e^{18 \cdot \pi \cdot i/6}
|
-23,722
|
3/7\times 3/8 = \dfrac{9}{56}
|
18,030
|
\left(x + 1\right)*3 = x*\left(2.5 + 3*(x + 1)\right) rightarrow x = 2/3
|
12,843
|
y^3 + (-1) = (1 + y^2 + y) \cdot ((-1) + y)
|
19,368
|
\cot(-x) = \dfrac{1}{\cos(-x)}\cdot \sin(-x) = (\left(-1\right)\cdot \sin(x))/\cos\left(x\right) = -\cot(x)
|
-1,501
|
\frac{1}{3}\times 7/(\frac{1}{7}\times 9) = 7/3\times 7/9
|
25,961
|
h^{x/\xi} h^{t/s} = h^{(x s + t \xi)/(\xi s)} = h^{x/\xi + t/s}
|
10,613
|
c \cdot b^{-k} = c \cdot b^{-k}
|
24,287
|
G^{k + 1}*y^{k + 1} = G^k*G*y^{k + 1} = \left(k + 1\right)*G^k*y^k
|
-10,364
|
\tfrac{1}{z \cdot 10} \cdot 4 \cdot 2/2 = \frac{8}{z \cdot 20}
|
443
|
x_{l + 1} = (1 + l) (1 + x_l) \implies 1 + l = \tfrac{x_{1 + l}}{1 + x_l}
|
4,544
|
(nx_2 + x_1 n a) z = nx_1 z a + nx_2 z
|
3,543
|
2*(-1) + z * z - z = (z + 1)*(2*(-1) + z)
|
47,671
|
71.5 = 1/2 \cdot 13 \cdot 11
|
-3,589
|
\dfrac{z}{z^4} \cdot \tfrac{3}{4} = \frac{3 \cdot z}{4 \cdot z^4}
|
-30,158
|
9\cdot z^8 = \frac{\mathrm{d}}{\mathrm{d}z} z^9
|
-20,297
|
\frac191 = \frac{1}{36 + 9r}(r + 4)
|
-22,340
|
x^2 - x*11 + 30 = (x + 6\left(-1\right)) (x + 5(-1))
|
21,386
|
3^2\cdot 3^n = 3^{n + 2}
|
23,487
|
2^f + 3^b = (1 + 1)^f + (1 + 2)^b \leq 1 + f + 1 + b \cdot 2
|
-12,446
|
\frac12\cdot 106 = 53
|
29,673
|
2 n + (2 n + 1)^3 + 1 = 8 n^3 + 12 n^2 + 8 n + 1 = 4*(2 n^3 + 3 n n + 2 n + 1)
|
28,156
|
\sqrt{2}/2 = \sqrt{2} \cdot 1/(\sqrt{2})/(\sqrt{2})
|
12,343
|
3 = (g + d \sqrt{2}) (g + d \sqrt{2}) = g^2 + 2 d^2 + 2 g d \sqrt{2}
|
38,154
|
1.6 = \tfrac{1}{10} \cdot 16
|
11,399
|
10 \cdot 10 \pi/2 = 50 \pi
|
-27,100
|
\sum_{n=1}^\infty (-2 + 3)^n/n = \sum_{n=1}^\infty \frac1n
|
16,200
|
x^4 + (-1) = \left(x + (-1)\right)*(x + 1)*\left(x^2 + 1\right)
|
-2,679
|
\sqrt{4 \cdot 3} + \sqrt{25 \cdot 3} = \sqrt{75} + \sqrt{12}
|
-20,005
|
\dfrac44 \frac{(-1)\cdot 6 p}{-p\cdot 10 + 3} = \frac{p\cdot (-24)}{-p\cdot 40 + 12}
|
3,227
|
|x| \cdot |Y| = |Y \cdot x|
|
8,066
|
3 * 3 + 4^2 + 5 * 5 = 5^2 + 5^2 = 2*5^2
|
-20,401
|
\frac{12 - s \cdot 40}{s \cdot 4 + 24 \cdot (-1)} = 4/4 \cdot \frac{1}{6 \cdot \left(-1\right) + s} \cdot \left(-10 \cdot s + 3\right)
|
16,629
|
z\cdot \sigma = \sigma\cdot z
|
4,630
|
\left(2 + \left(-1\right)\right)*(3 + (-1)) = 1*2 = 2
|
46,536
|
145 = 5\cdot 29
|
-7,094
|
2/5 = \frac{1}{5}\times 2
|
1,400
|
0 + B + 0 = \frac{1}{-2} \cdot (0 + 2) \cdot (0 + 5 \cdot (-1)) \implies 5 = B
|
43,042
|
4*(-1) + 4 + h = h
|
5,018
|
\frac{1}{\binom{n + (-1)}{2}}\cdot \binom{3\cdot \left(-1\right) + n}{2} = \frac{\left(n + 4\cdot (-1)\right)\cdot (n + 3\cdot (-1))}{(n + 2\cdot (-1))\cdot ((-1) + n)}
|
53,801
|
z^{3/2} \cdot (1 + 3/2 \cdot \frac{y}{z} + (-1) + \frac{3}{2} \cdot y/z) = z^{3/2} \cdot 3 \cdot y/z = 3 \cdot z^{\tfrac{1}{2}} \cdot y
|
12,054
|
2\cdot \sin{B}\cdot \sin{A} = \cos(A - B) - \cos(A + B)
|
16,260
|
(z - \psi_1)\cdot (z - \psi_2) = z^2 - z\cdot (\psi_2 + \psi_1) + \psi_2\cdot \psi_1
|
1,006
|
x + 3 - 4\sqrt{x + (-1)} = x + (-1) - 4\sqrt{x + \left(-1\right)} + 4 = \left(\sqrt{x + (-1)} + 2(-1)\right) * \left(\sqrt{x + (-1)} + 2(-1)\right)
|
3,106
|
\frac{1}{(y^2 + 1)^2} = -\frac{y^2}{\left(y^2 + 1\right)^2} + \frac{1}{y^2 + 1}
|
9,588
|
H = H \cap K rightarrow \{H,K\}
|
-6,692
|
1/100 + 60/100 = \frac{6}{10} + 1/100
|
28,146
|
(z*x)^2 = x^2*z^2
|
-4,971
|
0.54\cdot 10^{(-5)\cdot (-1) - 2} = 0.54\cdot 10^3
|
4,969
|
F = F^{\frac12}\cdot F^{\dfrac12}
|
20,491
|
\frac{1}{2^n\cdot 1/2} = \dfrac{1}{2^{\left(-1\right) + n}}
|
-27,337
|
\sin{z \cdot 2} = 2 \cdot \cos{z} \cdot \sin{z}
|
-20,140
|
\frac{9*z + 54}{27*(-1) + 81*z} = \frac{9}{9}*\dfrac{6 + z}{z*9 + 3*(-1)}
|
622
|
\frac{1}{1 + z} = \frac{1}{(\dfrac1z + 1) \cdot z}
|
8,035
|
1/r + \frac1q = 1 \implies 1 = ((-1) + r) \cdot ((-1) + q)
|
5,168
|
5 + 4 \cdot z^2 + z \cdot 4 = 4 \cdot (5/4 + z^2 + z)
|
-22,343
|
6 + p^2 + p\cdot 7 = (p + 1)\cdot (6 + p)
|
22,794
|
\left(-1\right)^{a - b} = (-1)^{a - b}\cdot \left(-1\right)^{2\cdot b} = \left(-1\right)^{a + b}
|
27,974
|
\sqrt{2 - 2\cdot \cos{B}} = 2\cdot \sin{\frac{1}{2}\cdot B} = 2\cdot \cos{\frac12\cdot (\pi - B)}
|
33,242
|
g^Y = (x \cdot F)^Y \Rightarrow F^Y \cdot x^Y = g^Y
|
33,030
|
\tan^{-1}{\dfrac{1}{\sqrt{2}}\cdot 0} = 0
|
42,944
|
x^{24} = x^{2 \cdot 9 + 2 \cdot 3} = x^9 \cdot x^9 \cdot x^3 \cdot x^2 \cdot x
|
40,162
|
4 + y = 3 + y + 1 = 3\cdot (1 + \frac13\cdot (y + 1))
|
-28,807
|
365 = \frac{2}{2 \times \pi \times \dfrac{1}{365}} \times \pi
|
39,196
|
100\cdot 102 + 2 = (101 + (-1))\cdot (101 + 1) + 2 = 101 \cdot 101 + (-1) + 2 = 101 \cdot 101 + 1
|
-4,299
|
\frac{55}{40} \cdot \tfrac{x^4}{x^4} = \frac{55}{40 \cdot x^4} \cdot x^4
|
18,077
|
\left(q\cdot \sin{t} = a + \cos{t}\cdot q\Longrightarrow a = (\sin{t} - \cos{t})\cdot q\right)\Longrightarrow (-\sin{t\cdot 2} + 1)\cdot q^2 = a \cdot a
|
20,445
|
x^2 + x + 1 = \dfrac{1}{x + (-1)}\times ((-1) + x^3)
|
-20,520
|
10/10\cdot (-\frac{6}{5}) = -\dfrac{1}{50}\cdot 60
|
24,312
|
2 x = 2 - x = 2 - x
|
-18,597
|
4\cdot y + 2\cdot (-1) = 6\cdot (y + 9\cdot (-1)) = 6\cdot y + 54\cdot \left(-1\right)
|
22,220
|
i k l i k l = i i k^2 l^2
|
6,932
|
2 \cdot b^2 + 2 \cdot a^2 = \left(a + b\right)^2 + (a - b)^2
|
-9,844
|
\frac{8}{25} = \frac{16}{50}
|
26,747
|
\dfrac{49}{48 \cdot 48} = \frac{1}{2304} \cdot 49
|
13,655
|
7 = x + \frac{1}{7}(-x^2 + 9x + 1) = (-x^2 + 16 x + 1)/7
|
88
|
\binom{2 + t}{2}\cdot 2 - \binom{1 + t}{1}\cdot 3 + \binom{0 + t}{0} = t \cdot t
|
-6,517
|
\frac{1}{(9 + y)\cdot (4\cdot (-1) + y)}\cdot 4 = \frac{4}{y \cdot y + 5\cdot y + 36\cdot \left(-1\right)}
|
21,772
|
\frac{h \cdot b_2}{x \cdot b_1} \cdot 1 = \frac{1/x \cdot h}{b_1 \cdot \frac{1}{b_2}}
|
17,331
|
z^4 - 7 \cdot z^2 + 1 = (z^2 + 1)^2 - 9 \cdot z \cdot z = (z^2 + 1 + 3 \cdot z) \cdot (z^2 + 1 - 3 \cdot z)
|
-28,765
|
-\frac{1}{4 + 2*z}*3 + \frac12 = \dfrac{1}{4 + z*2}*(z + \left(-1\right))
|
-4,335
|
\frac{60\cdot z}{50\cdot z^3} = 60/50\cdot \frac{z}{z^3}
|
19,895
|
(1 - \cos(A)) \times \left(\cos(A) + 1\right) = \left(1 - \cos(A)\right) \times \left(1 + \cos(A)\right) = 1 - \cos^2(A) = \sin^2\left(A\right)
|
31,170
|
2\cdot \frac{\sqrt{64}}{3} = 16/3
|
36,388
|
(1 + 2 + ... + k) (1 + 2 + ... + k) = 1^3 + 2^3 + ... + k^3
|
-20,298
|
\frac{1}{-k*10 + 9*(-1)}*3*\frac{1}{8}*8 = \frac{24}{72*(-1) - 80*k}
|
15,835
|
\pi + \arctan{-1} = \pi - \pi/4 = \frac34 \cdot \pi
|
38,209
|
z - -1004 = 1004 + z
|
24,136
|
e^{i\times y_m} = (-1)^m = e^{-i\times y_m}
|
30,530
|
\frac{\binom{10}{5}}{2^{11}} = 63/512
|
31,466
|
2 + \sqrt{3} = e^{-i \cdot z} = \cos{z} - i \cdot \sin{z}
|
8,834
|
|4 - \frac{9}{2}| = |5 - \frac{9}{2}|
|
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