id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
22,196 |
\mathbb{E}[(2\cdot A + 1) \cdot (2\cdot A + 1)] = \mathbb{E}[4\cdot A^2 + 4\cdot A + 1] = 4\cdot \mathbb{E}[A^2] + 4\cdot \mathbb{E}[A] + 1
|
34,316 |
\dfrac{1}{y} = \dfrac{1}{y}
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25,085 |
f^3 + b^3 + h^3 + (fb + hb + fh) (f + b + h) = (f + b + h)^3 \Rightarrow 0 = f^3 + b^3 + h * h^2
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34,843 |
\cosh x + \sinh x = e^x
|
956 |
\sin(90 - \theta) = -\cos{90} \sin{\theta} + \sin{90} \cos{\theta} \Rightarrow \cos{\theta}
|
-1,510 |
\frac{20}{12} = \frac{20 \cdot 1/4}{12 \cdot 1/4} = 5/3
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19,814 |
25^{x + 2} - 5^{2 + x*2} = ((-1) + 25)*5^{2 + 2*x}
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34,862 |
x^3 = y \cdot y \cdot y = z \cdot z^2 = x\cdot y\cdot z
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30,204 |
(-1) + x^4 = (x + 1)\cdot (x \cdot x + 1)\cdot ((-1) + x)
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7,941 |
F \times ( i \times x, w) = F \times \left( -x, i \times w\right) = -F \times \left( x, i \times w\right)
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-5,623 |
\frac{5}{s*3 + 12*(-1)} = \frac{1}{3*(4*(-1) + s)}*5
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1,231 |
8/45 = \frac13\cdot \frac{8}{15}
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-17,468 |
64 - 37 = 27
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1,937 |
(-x^2 + 1) \cdot (-x + 1)^2 = -x^4 + 2 \cdot x^3 - x \cdot 2 + 1
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13,306 |
\dfrac{1}{3^{1 / 2}} \cdot 75^{1 / 2} = \left(75/3\right)^{1 / 2} = 25^{\frac{1}{2}} = 5
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19,580 |
1 + h^2 + 3*h = 0 \Rightarrow h^2 = -h*3 + (-1)
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26,820 |
\left((-6) + E \cdot 9 = 3 \Rightarrow 3 + 6 = E \cdot 9\right) \Rightarrow E = 1
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14,534 |
h - a = |h - a| < h - a
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20,728 |
(b - c)^2 \geq 4(1 + b + c)^2 - 4(b + c) * (b + c) = 4*(1 + 2b + 2c) > 8(b + c)
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-10,291 |
16 = -10 + 20*m + 40*(-1) = 20*m + 50*(-1)
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-29,627 |
\frac{\mathrm{d}}{\mathrm{d}x} (2*x^4 + x^3 + 3*x * x) = x*6 + x^3*8 + 3*x^2
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10,427 |
\left(-1\right) + A^n = (A + (-1))*(1 + A + \ldots + A^{n + (-1)})
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-20,492 |
3/3 \cdot \frac{1}{r + 6 \cdot \left(-1\right)} \cdot (r + 10 \cdot (-1)) = \frac{r \cdot 3 + 30 \cdot (-1)}{3 \cdot r + 18 \cdot \left(-1\right)}
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10,139 |
75\cdot \dfrac{1}{180}\cdot \pi = \frac{5}{12}\cdot \pi
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27,670 |
\cot{Z} - \tan{Z} = (\cos^2{Z} - \sin^2{Z})/\left(\sin{Z}\cdot \cos{Z}\right) = 2\cdot \cot{2\cdot Z}
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26,691 |
a^4 + a^2 = (a^2 + a)^2 = (a*\left(a + 1\right))^2 = (a*a^4)^2 = a^{10}
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-6,393 |
\frac{4}{(6 + x)*3} = \dfrac{1}{18 + 3*x}*4
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21,051 |
\frac{dp}{ds} = \frac{4p^2}{4ps} = \dfrac1sp
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35,669 |
x_0\cdot f = f\cdot x_0
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-3,246 |
2 \sqrt{5} + \sqrt{5} = \sqrt{5} + \sqrt{4} \sqrt{5}
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-30,565 |
-\tfrac{243}{-81} = -81/(-27) = -\frac{1}{-9}27 = 3
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-16,779 |
-5 = 12 \cdot l^2 - 15 \cdot l - 5 \cdot 4 \cdot l - -25 = 12 \cdot l^2 - 15 \cdot l - 20 \cdot l + 25
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3,138 |
e \geq 2 \cdot (2 - l + e) \implies 4 \cdot (-1) + l \cdot 2 \geq e
|
20,791 |
(x^2*2 + 2y^2) (2yy' + 2x) = 4y * y * y y' + x^3*4 + 4y' y x * x + 4y^2 x
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-15,620 |
\dfrac{r^{16}\cdot \frac{1}{z^8}}{\frac{1}{r^5}\cdot \frac{1}{z^5}}\cdot 1 = \tfrac{r^{16}}{\frac{1}{r^5}}\cdot \dfrac{1}{z^8\cdot \frac{1}{z^5}} = \frac{1}{z^3}\cdot r^{16 - -5} = \tfrac{r^{21}}{z^3}
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-7,381 |
\dfrac{1}{13}\times 4\times \frac{1}{14}\times 5 = \frac{10}{91}
|
-3,919 |
\dfrac{5}{6} = \frac{1}{6}5
|
20,155 |
(1^2 + 1 \cdot 1 + 2^2)^2 - (1^3 + 1^3 + 2^3)\cdot (1 + 1 + 2) = 36 + 40\cdot (-1) = -4
|
30,287 |
(z + \sqrt{1 - z^2})^2 = z^2 + 2 \cdot z \cdot \sqrt{1 - z^2} + 1 - z^2 = 2 \cdot z \cdot \sqrt{1 - z^2} + 1
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24,690 |
\tfrac13 = 1/6 + 1/6
|
11,825 |
-x \cdot 2 + x^2 = \left(x + \left(-1\right)\right)^2 + (-1)
|
20,262 |
\cos{x} = \cos^2{x/2}\times 2 + \left(-1\right)
|
6,258 |
\frac{5}{16} = 1/8\cdot 20/8
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19,676 |
2 \cdot x + c_2 \cdot 4 = 0 rightarrow x = -2 \cdot c_2
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35,378 |
\frac{1}{a \cdot 1/b} = \frac{1}{a} \cdot b
|
-4,631 |
\dfrac{-4x + 19}{x^2 - 9x + 20} = -\frac{1}{x + 5(-1)} - \frac{3}{x + 4(-1)}
|
22,389 |
2\cdot \delta\cdot x + \delta^2 = (\delta + x)^2 - x^2
|
30,914 |
\cos(\phi) = -2*\sin^2(\phi/2) + 1
|
2,070 |
(95 + 12 \cdot \left(-1\right)) \cdot (12 + 95) = 8881
|
4,050 |
\operatorname{re}{(e^{i*x} * e^{i*x})} = \operatorname{re}{\left(e^{2*x*i}\right)} \Rightarrow \cos(x*2) = \cos^2(x) - \sin^2(x)
|
26,911 |
4 = z\cdot 2 - y \Rightarrow y + 4 = 2\cdot z
|
10,723 |
2 \times z \times w + w^2 + 3 = 0 = z^2 + 2 \times z \times w
|
9,496 |
X^2 + Y^2 = X^2 + Y^2 - 2*X*Y + 2*X*Y = (X - Y) * (X - Y) + 2*X*Y
|
22,415 |
x^{\dfrac{1}{2}} = x^{\tfrac{1}{2}}
|
30,808 |
y^3 - y^2 + 2*y + (-1) = y + 2*y^2 + 2*y + 2 = 2*y * y + 2
|
3,411 |
\dfrac{\pi}{8} = -\pi/16 + 3 \cdot \pi/16
|
14,692 |
\|v\|^2 = (\sqrt{v_1 \cdot v_1 + v_2^2 + v_3^2})^2 = v_1^2 + v_2^2 + v_3^2
|
20,751 |
T^{\frac{1}{2}} \cdot x^{1/2} \cdot T^{1/2} \cdot x^{1/2} = T \cdot x
|
12,561 |
h^3 + d * d * d + x^3 - 3*d*x*h = \left(-h*x + h^2 + d^2 + x^2 - d*h - x*d\right)*(x + h + d)
|
31,889 |
\tan^{-1}\left(1\right) + \tan^{-1}(2) + \tan^{-1}(3) = \tan^{-1}\left(-3\right) + \tan^{-1}(3) = 0
|
24,119 |
\sqrt{z + 2}*(z + 2) = (z + 2)^{3/2}
|
1,734 |
2^{1 + k} + (-1) = 2^{k + 1} + (-1)
|
4,720 |
z^3 + z \cdot z = 2\cdot z - z^2 = 2\cdot z - 2 - 2\cdot z = 4\cdot z + 2\cdot (-1)
|
4,293 |
\tan(x*g) = \tan(0.5*x*g + 0.5*x*g) = \frac{\tan(0.5*x*g)}{1 - \tan^2(0.5*x*g)}*2
|
27,305 |
10836 = 3^2\cdot 2^2\cdot 7\cdot 43
|
19,496 |
(-C + A) \left(C + A\right) = -C * C + A^2
|
21,503 |
V \gt 1 - U \implies U \gt 1 - V
|
19,929 |
\left(0^X\right)^C = 0^C = 1 = 1^X = (0^C)^X
|
-5,817 |
\frac{1}{(6(-1) + q) (q + 10)}4 = \frac{4}{q^2 + 4q + 60 (-1)}
|
27,447 |
\frac{\frac{1}{19^{20}}}{\frac{1}{20^{20}}} = \tfrac{1}{19^{20}}20^{20} \approx 2.79
|
-20,066 |
\frac13 3 \dfrac{1}{\left(-8\right) z} (-5 z + 2 (-1)) = \dfrac{1}{\left(-24\right) z} (-15 z + 6 (-1))
|
24,743 |
x \cdot l + x = \left(l + 1\right) \cdot x
|
15,231 |
v = \tan^{-1}(z_2/(z_1)) \Rightarrow \tan(v)\cdot z_1 = z_2
|
-20,978 |
\frac{-4\cdot t + 5}{4 - t\cdot 7}\cdot 3/3 = \frac{-t\cdot 12 + 15}{12 - 21\cdot t}
|
10,596 |
\dfrac{1}{x \cdot y} = \tfrac{1}{y \cdot x}
|
22,105 |
\mathbb{N} \coloneqq \left\{2, \dots, 3, 1\right\}
|
28,443 |
\left|{x}\right| \left|{-Y + \lambda \nu}\right| \left|{1/x}\right| = \left|{x}\right| \left|{\nu \lambda - Y}\right| \left|{1/x}\right|
|
-9,464 |
-6 r = -r*2*3
|
15,419 |
(n*y + d - \eta)*\left(n*y + d - \eta\right) = (-\eta + y*n + d)^2
|
-5,773 |
\frac{1}{3 \times (o + 9)} = \frac{1}{27 + 3 \times o}
|
-4,899 |
0.78\times 10^{2 + 0\times \left(-1\right)} = 10^2\times 0.78
|
35,737 |
\cot(1) \sin\left(1\right) = \cos(1)
|
11,268 |
4\cdot j + 7 = 2\cdot (2\cdot j + 3) + 1
|
8,392 |
x^4 + (-1) = (1 - x)\cdot (1 + x)\cdot ((-1) - x^2)
|
10,464 |
ma xh' = ah' \frac{ax}{a}1 m
|
18,397 |
2^{l + 1} = 2\cdot 2^l \geq 2\cdot 4\cdot l \cdot l > 4\cdot l^2 + 8\cdot l + 4 = 4\cdot (l \cdot l + 2\cdot l + 1) = 4\cdot (l + 1)^2
|
-1,715 |
\pi*\frac{13}{6} = \pi*\frac43 + \pi*\frac{5}{6}
|
-30,859 |
\frac{12}{\left(-1\right) + z} = \frac{1}{-z^2 + z^4}(z^3*12 + 12 z^2)
|
392 |
1/4 = -\dfrac{11}{16} + \frac{5}{16}\cdot 3
|
15,872 |
Z = \frac14X + Z/2 \implies Z = X/2
|
28,335 |
\left(d^2 \sinh^2(t) + d \cdot d\right)^{1/2} = d \cdot (\sinh^2(t) + 1)^{1/2} = d\cosh(t)
|
18,568 |
x^2 + y \cdot y + z^2 + 2\cdot (z\cdot x + y\cdot x + z\cdot y) = 36\Longrightarrow 9 = z\cdot x + y\cdot x + y\cdot z
|
35,516 |
\frac{1}{13}13 = 1
|
12,174 |
\tfrac34 = \frac12 + 1/4
|
4,966 |
-\tfrac{1}{(-1) + y_{l + 1}} + 1 = \dfrac{y_{l + 1} + 2\cdot (-1)}{\left(-1\right) + y_{l + 1}}
|
24,018 |
\cos(3\cdot x) = -\cos(x)\cdot 3 + 4\cdot \cos^3(x)
|
7,288 |
\frac{1}{\binom{4}{2}} \times (-\binom{2}{2} + \binom{3}{2}) = \frac{1}{3}
|
20,657 |
\dfrac{c}{4 \cdot a} = \frac{c \cdot c}{4 \cdot a \cdot c} \geq \frac{1}{(c + a)^2} \cdot c^2
|
24,739 |
g_1\cdot \dotsm\cdot g_\delta = a\cdot g_1\cdot \dotsm\cdot a\cdot g_\delta = a^\delta\cdot g_1\cdot \dotsm\cdot g_\delta
|
27,504 |
\tfrac{1}{2}*2*2 = 2 = 1/2*0 = 0
|
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