id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
12,618 |
\sqrt{x - g} \dfrac{x - g - x - h}{\left(-h + x\right) (-g + x)} = \dfrac{-g + h}{\sqrt{-g + x} (x - h)}
|
4,392 |
(a^2 - g^2) \cdot u = (a + g) \cdot (-g + a) \cdot u
|
10,543 |
x^2 = -(x + 3 \times (-1)) \times (-x + 5) - 4 \times (8 - 2 \times x) + 17
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8,295 |
\left(6 + 20^{1 / 2}\right)^{1 / 2} = \left(5^{1 / 2} \cdot 2 + 6\right)^{\dfrac{1}{2}}
|
12,995 |
(c - d)^2 = d^2 + c^2 - cd \cdot 2
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33,375 |
(2\cdot (-1) + z\cdot 3)\cdot (5 + z) = 10\cdot (-1) + z^2\cdot 3 + 13\cdot z
|
-657 |
(e^{\dfrac{7}{4} \cdot i \cdot \pi})^3 = e^{3 \cdot \frac14 \cdot \pi \cdot 7 \cdot i}
|
-9,092 |
138\% = \frac{138}{100}
|
-7,426 |
9/91 = \frac{6}{14} \cdot \frac{3}{13}
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-6,707 |
\frac{1}{100}\cdot 30 + \dfrac{5}{100} = \frac{5}{100} + \frac{3}{10}
|
36,087 |
17 + 4 \cdot (-4) = 1
|
-10,385 |
\frac{40 + s \cdot 4}{60 \cdot s + 40 \cdot (-1)} = \frac{s + 10}{10 \cdot (-1) + 15 \cdot s} \cdot \frac{1}{4} \cdot 4
|
-7,627 |
(-2 + 26 i - i + 13 (-1))/5 = (-15 + 25 i)/5 = -3 + 5i
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31,459 |
\int\limits_g^f \cot{z}\,\mathrm{d}z = \int\limits_g^f \cot{z}\,\mathrm{d}z
|
21,892 |
(2 + (-1) + 2\cdot x + (-1))/2 = x
|
23,425 |
2\cdot \sin{u}\cdot \sin{v} = \cos(-u + v) - \cos(v + u)
|
36,075 |
-\frac{1}{27} = \dfrac{1}{-27}
|
7,351 |
\mathbb{Var}\left(Q\right) = \mathbb{E}\left((Q - \mathbb{E}\left(Q\right))^2\right) = \mathbb{E}\left(Q^2\right) - \mathbb{E}\left(Q\right)^2
|
-14,002 |
10\cdot 7 + 2\cdot 24/3 = 10\cdot 7 + 2\cdot 8 = 70 + 2\cdot 8 = 70 + 16 = 86
|
24,872 |
3 = y\cdot 4 - 2\cdot (1 - y) \Rightarrow 5/6 = y
|
36,721 |
\sqrt{f'^2 + f \cdot f} = \sqrt{f^2 + f'^2}
|
12,149 |
\frac{-x + 1}{3\times (x^2 + 2)} + \frac{1}{3\times \left(1 + x\right)} = \dfrac{1}{(x \times x + 2)\times (x + 1)}
|
15,390 |
\sqrt{0.5 \cdot 2} = 1
|
-25,807 |
\tfrac{2}{7\cdot 6} = 2/42
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-25,689 |
d/dz (\frac{4}{z + 2}) = -\frac{4}{(2 + z)^2}
|
-11,563 |
-4 + 15 (-1) + i*4 = i*4 - 19
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-18,358 |
\dfrac{(6*(-1) + t)*t}{(1 + t)*\left(t + 6*(-1)\right)} = \frac{-t*6 + t^2}{t^2 - 5*t + 6*\left(-1\right)}
|
-18,403 |
\frac{f^2 + f}{f^2 - 3*f + 4*(-1)} = \tfrac{1}{(f + 1)*(f + 4*(-1))}*f*(1 + f)
|
-22,315 |
(9 + s)\cdot (4 + s) = s^2 + s\cdot 13 + 36
|
-6,049 |
\frac{y\cdot 8}{8\cdot (y + 6\cdot (-1))\cdot (y + 9\cdot (-1))} = \frac{y}{(6\cdot (-1) + y)\cdot (y + 9\cdot (-1))}\cdot \frac{1}{8}\cdot 8
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33,138 |
D_x\cdot D_i = D_x\cdot D_i
|
53,788 |
\frac{(n-1)^{n-1}}{n^{n-2}} = (n-1)\left(1-\frac{1}{n}\right)^{n-2} = (n-1)\left(1-\frac{1}{n}\right)^n \left(1-\frac{1}{n}\right)^{-2} \\ = \frac{n^2}{n-1} \left(1-\frac{1}{n}\right)^n
|
2,974 |
(1 + 6)\times (4 + 1)\times (2 + 1) = 105
|
-1,715 |
\pi*5/6 + \pi \tfrac134 = \pi*13/6
|
8,189 |
1 = z^3 + z - y^2 \implies z^3 + z + (-1) = y * y
|
6,658 |
(-1) + 2 \cdot \cos^2{Z} = \cos{2 \cdot Z}
|
16,784 |
I' + I*\lambda = \left(I' + \lambda*I\right)^2
|
4,828 |
-\frac{\pi}{4} \cdot 7 = -\dfrac{\pi}{2} \cdot 3 - \frac{1}{4} \cdot \pi
|
8,189 |
z^3 + z - y \cdot y = 1 \Rightarrow (-1) + z^3 + z = y^2
|
32,915 |
\eta^z + \left(-1\right) = (\eta + (-1))\times (\eta^{(-1) + z} + \eta^{2\times (-1) + z} + ... + \eta + 1)
|
-17,142 |
2 = 2\cdot 2\cdot q + 2\cdot (-6) = 4\cdot q - 12 = 4\cdot q + 12\cdot (-1)
|
25,426 |
(1 + n)^4 = 1 + n^4 + n * n^2*4 + 6n * n + n*4
|
587 |
\cot{y} - 8 \cdot \cot{8 \cdot y} = \cot{y} - 8 \cdot \cot{8 \cdot y} = \cot{y} - 8 \cdot \dfrac{1}{2 \cdot \cot{4 \cdot y}} \cdot (\cot^{24}{y} + (-1))
|
-16,624 |
8 = 8\cdot (-5\cdot t) + 8\cdot (-1) = -40\cdot t - 8 = -40\cdot t + 8\cdot (-1)
|
3,329 |
10^{n + m} = 10^n \cdot 10^m
|
17,710 |
|c h| = |c| |h|
|
16,299 |
x_1 + x_2 + v_1 + v_2 = v_2 + x_2 + v_1 + x_1
|
3,670 |
\binom{4}{2} = \tfrac{4!}{2!*(4 + 2*(-1))!} = 6
|
23,089 |
2^{1 + k} - 1 + 2^k + 1 = 2^k
|
7,858 |
(t + t_2 + t_1)^2 - t^2 + t_2^2 + t_1^2 = 2\cdot (t_2\cdot t_1 + t\cdot t_1 + t\cdot t_2)
|
21,772 |
\frac{xg}{hf}1 = \dfrac{\dfrac1h g}{1/x f}1
|
13,052 |
\frac{5}{13} = (10*9 + 2*10)/(\binom{13}{3})
|
28,002 |
y = z + 2 \Rightarrow y + 2 \left(-1\right) = z
|
2,223 |
-(4 + 1 + 2 + 3) = 16 \cdot (-1) + 1 + 4 \cdot (-1) + 9
|
-22,153 |
30/50 = \dfrac15\cdot 3
|
17,559 |
2^{k + 3} + 2^{2 + k} + 2^{1 + k} + 2^k + 2^k = 2^{k + 4}
|
9,215 |
1 + \frac{1}{k \cdot k} - 2/k = (1 - 1/k) \cdot (1 - 1/k)
|
11,422 |
3 \cdot \left(5 \cdot (-1) + 2 \cdot y\right)^2 \cdot 2 = \left(2 \cdot y + 5 \cdot \left(-1\right)\right) \cdot \left(2 \cdot y + 5 \cdot \left(-1\right)\right) \cdot 6
|
6,924 |
l_k\times p_k = l_k\times p_k
|
-2,313 |
\dfrac{8}{19} - \dfrac{2}{19} = \dfrac{6}{19}
|
36,915 |
\alpha_x = \alpha_x
|
13,049 |
-4\cdot (1 + z^2) + 4 = -4\cdot z^2
|
3,942 |
|z_2 + z_1 + \varphi| = |z_2 + z_1 + \varphi|
|
-5,491 |
\frac{z*2}{(z + 1) (z + 2(-1))} = \frac{2z}{z^2 - z + 2(-1)}
|
7,847 |
7(-100) + 11 \cdot 67 = 37
|
-30,291 |
\tfrac12\cdot (0 + 4) = 4/2 = 2
|
-12,810 |
7 = 15 + 8 (-1)
|
39,283 |
3*(12 + 6 (-1)) = 18
|
-7,618 |
\frac{-6 + i\cdot 8}{-1 - 2\cdot i} = \tfrac{-1 + 2\cdot i}{-1 + i\cdot 2}\cdot \frac{i\cdot 8 - 6}{-2\cdot i - 1}
|
11,449 |
-\sin{π/6} - -\sin{\frac56\cdot π} = 0
|
-11,999 |
4/15 = \frac{s}{6\pi}*6\pi = s
|
24,838 |
\mathbb{E}[Y \cdot V] = \mathbb{E}[Y] \cdot \mathbb{E}[V]
|
15,441 |
z^{50} = z^{16} \cdot z^2 \cdot z^{32}
|
31,468 |
e^{-i y} = \cos(-y) + i \sin(-y) = \cos(y) - i \sin(y)
|
25,876 |
-\left(2 (-1) + x_n^2\right)/(x_n*2) + x_n = \frac12 \left(x_n + 2/\left(x_n\right)\right)
|
-9,925 |
0.82 = 8.2/10 = \dfrac{1}{50} \cdot 41
|
11,396 |
20/4 = 2^1*5/2
|
21,075 |
\frac{8}{24} - 21/24 = \left(8 + 21\times (-1)\right)/24 = -13/24
|
19,134 |
280 = (20 \cdot \left(-1\right) + 120) \cdot 3 + 20 \cdot (-1)
|
4,552 |
\dfrac{1}{x^3}(x^3 - 7x) = 1 - \frac{7}{x \cdot x}
|
31,466 |
2 + \sqrt{3} = e^{-i*x} = \cos(x) - i*\sin(x)
|
55,574 |
10 = \binom{5}{3}
|
-22,804 |
\frac{54}{4 \cdot 18} \cdot 1 = 54/72
|
34,545 |
\dfrac{p}{n} \cdot \tfrac{p + \left(-1\right)}{n + (-1)} = \tfrac{1}{-n + n^2} \cdot (p^2 - p)
|
-22,199 |
(9 \cdot (-1) + q) \cdot (6 + q) = q \cdot q - q \cdot 3 + 54 \cdot (-1)
|
18,918 |
\frac{z^2}{z^2 + (-1)}*2 = 2 + \dfrac{1}{z^2 + (-1)}*(z + 1 - z + (-1)) = 2 + \frac{1}{z + (-1)} - \dfrac{1}{z + 1}
|
1,704 |
u \cdot D^1 = D^{\frac12} \cdot u \cdot D^{\dfrac{1}{2}}
|
-6,689 |
0/100 + \frac{1}{100}*7 = 0/10 + \frac{1}{100}*7
|
16,162 |
\frac{42}{2}1 + 14 \left(-1\right) = 7 < 14
|
10,012 |
\dfrac{x^{4i}}{A_i^4} = x^l \Rightarrow x^{i\cdot 4 - l} = A_i^4
|
-11,595 |
-2 + 4 - 9*i = 2 - 9*i
|
-17,837 |
8\left(-1\right) + 39 = 31
|
-11,611 |
-12 + 12 i = 0 + 12 (-1) + 12 i
|
-3,810 |
2\times 3\times 7 = 42
|
26,721 |
13/27 = \frac{1}{3} + \dfrac{2}{9} \cdot 2/3
|
-11,794 |
\frac{1}{16} \cdot 9 = \left(3/4\right)^2
|
51,352 |
\lim_{n \to \infty} \|\frac{\left(n + 1\right)!}{(n + 1)^{n + 1}} n^n/n!\| = \lim_{n \to \infty} \|\frac{\left(n + 1\right) n!}{(n + 1)^{n + 1}} n^n/n!\| = \lim_{n \to \infty} \|\frac{n + 1}{\left(n + 1\right)^{n + 1}} n^n\|
|
53,756 |
2^{2012} = (2^4)^{503} = (-1 + 17)^{503}
|
-1,247 |
\dfrac{30}{35} = \dfrac{30 \times 1/5}{35 \times \frac{1}{5}} = 6/7
|
29,554 |
3^{n + 1} \gt 3n = n + 2n > n + 1
|
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