id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
10,817 |
r\cdot (1 + r + r \cdot r + r^3 + r^4 + r^5)/6 = (r + r^2 + r^3 + r^4 + r^5 + r^6)/6
|
25,950 |
7(-1) - \frac{1}{3}(1/5 (-24)) = -27/5
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17,090 |
7/2 = \tfrac{1}{1 + \sqrt{1 + 0}}\cdot 7
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18,703 |
(1 + x) \cdot m = m \cdot x + m
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16,900 |
\frac11 \cdot ((- 3 / 4) \cdot t^2) = -t^2 \cdot 3/4
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8,838 |
10^{-2\cdot x + 2 - -3 + x} = 10^{5 - 3\cdot x} = 10^{-4\cdot x + 2\cdot \left(-1\right)}
|
27,020 |
\sin{\frac1x} = \sin{1/x}
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26,904 |
8 + 4 + 4 + 4 + 2 \cdot (-1) + 2 \cdot (-1) - (-1) + 2 + 2 + 2 = 11
|
-1,201 |
-\tfrac{21}{24} = \dfrac{1}{24 \cdot 1/3} \cdot \left((-21) \cdot 1/3\right) = -\frac18 \cdot 7
|
8,333 |
z^2 - l^2 = (z + l)*\left(z - l\right)
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25,488 |
(a_l + 1)\cdot ((-1) + a_l) = (-1) + a_l^2
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34,789 |
1 + y^4 + y^2 = (y^2 - y + 1) \cdot \left(y \cdot y + y + 1\right)
|
10,743 |
y^2 \cdot (-\frac12) = (y^2 \cdot \left(-1\right))/2
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1,213 |
\dfrac{1}{1 - \cos(x)} \cdot \sin^2(x) = \dfrac{2}{\sin(x)} \cdot \sin\left(x\right) \cdot \cos(x) = 2 \cdot \cos(x)
|
34,855 |
z^{\dfrac{1}{m}} = z^{\dfrac{1}{m}}
|
3,037 |
\frac{\sin(y^5)}{y} = \frac{\sin(y^5)}{y^5} y^4
|
16,140 |
12 = 33 + 21 \cdot \left(-1\right)
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45,826 |
3!*4!*5 = 3!*5!
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32,919 |
\frac{1}{x} + 1/\epsilon + \dfrac{1}{M} = \frac{1}{\epsilon\times M\times x}\times (\epsilon\times x + \epsilon\times M + M\times x)
|
1,347 |
2^{k + \left(-1\right)} = 2^k/2
|
17,074 |
17^2 + 5 \cdot 5 + 7^2 + 11^2 + 13^2 = 653
|
33,982 |
0 \leq x \implies |x| = x
|
34,073 |
T\cdot C = T\cdot C
|
26,127 |
D*C + C*G = \left(D + G\right)*C
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26,828 |
83^{\frac{1}{2}} = (\left(-3\right)^2 + (-2) \cdot (-2) + \dots + 5^2)^{\frac{1}{2}}
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33,336 |
\frac{625}{(t + 5\cdot (-1))\cdot 9} - \frac{1}{9\cdot \left(t + 4\right)}\cdot 256 = \frac{1}{t^2 - t + 20\cdot (-1)}\cdot (420 + 41\cdot t)
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20,539 |
x^2 + x^2 + x * x = x*x + x*x + x*x + x*x*x + 1 = x^2 + x^2 + x * x + x^3 + 1
|
-727 |
e^{19\cdot 7\pi i/4} = (e^{\dfrac{i\pi\cdot 7}{4}})^{19}
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30,221 |
C = B \cup C \setminus B rightarrow \{B,C\}
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25,663 |
\left(b\cdot (-1)\right)/r = -\frac{b}{r}
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6,253 |
\dfrac{1}{\sqrt{-x^2 + 1}} = \frac{\mathrm{d}}{\mathrm{d}x} \sin^{-1}(x)
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-3,012 |
\sqrt{10} \cdot 5 + \sqrt{10} \cdot 3 = \sqrt{10} \cdot \sqrt{25} + \sqrt{10} \cdot \sqrt{9}
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-1,504 |
\dfrac{36}{72} = 36*\frac{1}{36}/\left(72*\dfrac{1}{36}\right) = \frac{1}{2}
|
4,719 |
a^3 - b \cdot b^2 = (a \cdot a + a\cdot b + b \cdot b)\cdot \left(a - b\right)
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31,275 |
(4 w + 6)/4 = w + \tfrac64 = w + 1 + 1/2
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4,085 |
1/(x \mu) = 1/(x \mu)
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5,129 |
y + 3 - 4\sqrt{y + (-1)} = y + (-1) + 4 - 4\sqrt{y + (-1)} = y + (-1) - 4\sqrt{y + (-1)} + 4 = (\sqrt{y + \left(-1\right)} + 2(-1))^2
|
-25,057 |
3/7 \cdot 3/8 = \dfrac{9}{56}
|
4,851 |
\sum_{i=1}^n i\cdot z^i = z\cdot \frac{\partial}{\partial z} \sum_{i=1}^n z^i
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12,207 |
-\frac{2}{1 + z \cdot 2} = -\frac{1}{2 \cdot z + 1} \cdot 2
|
-528 |
(e^{i \cdot \pi \cdot 11/6})^4 = e^{\frac{11}{6} \cdot \pi \cdot i \cdot 4}
|
-5,815 |
\frac{4}{2 \times (7 + k)} = \frac{4}{14 + 2 \times k}
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21,456 |
\dfrac{1}{1 + 0*(-1)} = 1
|
16,625 |
D^{x + l} = D^l\cdot D^x
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-20,576 |
\tfrac{-16\cdot n + 16}{10\cdot (-1) + n\cdot 10} = \dfrac{2\cdot (-1) + n\cdot 2}{n\cdot 2 + 2\cdot (-1)}\cdot \left(-\frac15\cdot 8\right)
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30,574 |
26^3 = (6 \cdot (-1) + 2^5)^3
|
23,701 |
c*2 + 1 = (1 + c)^2 - c^2
|
504 |
\frac{\pi}{3} - \pi/2 = -\pi/2 + \tfrac{\pi}{3}
|
15,434 |
d/dx y^2 = d/dy y \cdot y\cdot \frac{dy}{dx} = 2\cdot y\cdot \frac{dy}{dx}
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21,753 |
2 = \left(3 * 3 + 0^2 + 0^2 + 1^2 + 1^2 + 1^2\right)/6
|
-16,587 |
4 \cdot \sqrt{25 \cdot 11} = \sqrt{275} \cdot 4
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29,308 |
\overline{x*b} + \overline{b*x} = 10*x + b + 10*b + x = 11*(x + b)
|
42,685 |
\binom{6 + 2(-1)}{1} = \binom{4}{1} = 4
|
16,444 |
1/3 + \frac15 + 1/6 = 7/10 < \dfrac{3}{4}
|
27,537 |
1/(g*f) = \frac{1}{f*g}
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24,206 |
x^2+y^2+0\cdot y=x^2+y^2
|
1,538 |
\cos(c)*\sin(c)*2 = \sin(2*c)
|
10,295 |
4 + 27\cdot 2 + 81\cdot q = 58 + 81\cdot q
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29,840 |
z^2 - y^2 = \left(z - y\right)\cdot \left(y + z\right)
|
-5,446 |
\tfrac{0.86}{10} = 0.86/10
|
14,505 |
216 = 6 \cdot 6 \cdot 6 = 2^3\cdot 3^3
|
-1,577 |
-2*\pi + \frac{25}{12}*\pi = \frac{\pi}{12}
|
15,168 |
b_j + d_j = b_j + d_j
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15,290 |
\left(b \cdot 2 + 25 = 6 + a \Leftrightarrow (a + 6) \cdot 13 = 13 \cdot (b \cdot 2 + 25)\right) \Rightarrow a = 19 + b \cdot 2
|
2,341 |
t \cdot \cot{t} = 1 - t^2/3 - \frac{t^4}{45 \cdot \dotsm} \approx e^{((-1) \cdot t^2)/3} \cdot \left(1 - \frac{7}{90 \cdot t^4} + \dotsm\right)
|
-30,339 |
4 = 9 + 5*(-1)
|
25,555 |
y^2\times 9 + x^2 + 4\times x\times y = y^2\times 5 + (2\times y + x) \times (2\times y + x)
|
-20,892 |
-\frac{30}{25} = \frac{5}{5} \cdot (-\frac{6}{5})
|
4,094 |
\binom{l}{k} = 0 = \binom{l + \left(-1\right)}{k + (-1)} + \binom{l + \left(-1\right)}{k}
|
13,970 |
z^{30} + 1 = 1 + \left(z^2\right)^{15}
|
9,194 |
1 = v'' x^2 + xv' \Rightarrow v' + v'' x = 1/x
|
4,255 |
y^9 + \left(-1\right) = \left(y * y * y + (-1)\right)*\left(y^6 + y^3 + 1\right) = (y + (-1))*\left(y * y + y + 1\right)*\left(y^6 + y^3 + 1\right)
|
15,403 |
z + y \cdot z^2/2! + \cdots = \frac1y \cdot (\left(-1\right) + e^{y \cdot z})
|
8,873 |
{i + n \choose i} = {n + 1 + i \choose i} - {i + n \choose i + \left(-1\right)}
|
20,112 |
x \lt b \Rightarrow x^2 \lt b^2 = \frac{1}{9}
|
13,012 |
-6 = 1 + 1 + 8\left(-1\right)
|
-20,938 |
\frac{1}{81}\cdot (54 + 9\cdot x) = (6 + x)/9\cdot 9/9
|
4,369 |
(\dfrac{1}{2} \cdot \left(\sqrt{13} + 1\right)) \cdot (\dfrac{1}{2} \cdot \left(\sqrt{13} + 1\right)) \cdot (\dfrac{1}{2} \cdot \left(\sqrt{13} + 1\right)) = 5 + 2 \cdot \sqrt{13}
|
21,207 |
0 \lt \sqrt{7 - C} - \sqrt{7 - h} = \frac{h - C}{\sqrt{7 - C} + \sqrt{7 - h}} \lt \dfrac12\cdot (h - C)
|
29,793 |
\mathbb{Var}\left(X\right) = Cov\left(X,X\right)
|
3,862 |
(X + Y)^{x + 1} = \left(X + Y\right)^x*(X + Y) = (X + Y)^x X + \left(X + Y\right)^x Y
|
20,334 |
0.066^2\cdot 0.023^2\cdot \dotsm = \frac{1}{1000000000000000000000000000000000}\cdot 8.4
|
23,046 |
\left(5/3\right)^k = \frac{1}{3^k \cdot \frac{1}{5^k}}
|
194 |
4 \cdot x \cdot S = -(x - S)^2 + \left(S + x\right) \cdot \left(S + x\right)
|
6,078 |
\dfrac{121}{11\cdot (10 + 1)} = \frac{1}{11\cdot 11}\cdot 121 = 121/121 = 1
|
37,782 |
\pi = \arccos{-1}
|
-30,912 |
\frac{1}{50}b = b/150 \cdot 3
|
2,967 |
(1 - x) \cdot (1 - b) = 1 - x - b + x \cdot b \geq 1 - x - b
|
-12,908 |
5/8 = \dfrac{15}{24}
|
8,228 |
b'^2*\cos^2(x) + b'^2*\sin^2(x) + 2*b'^2*\cos(x) = -2*\cos(-x + \pi)*b' * b' + b' * b'
|
114 |
\frac{1/3*1/4}{2} = 1/24
|
8,636 |
a\cdot E\cdot n\cdot E/(a\cdot E) = n\cdot E = a\cdot \frac{n}{a}\cdot E
|
16,808 |
\left(y + 2\right)*(1 + y) = \frac{1}{y + 1}*\left(y^3 + y^2*4 + y*5 + 2\right)
|
10,855 |
a^{i + 2}\times b^{i + 2} = (a\times b)^{i + 2} = \left(a\times b\right)^{i + 1}\times a\times b = a^{i + 1}\times b^{i + 1}\times a\times b
|
20,959 |
(h + 3)^2 = h^2 + 3^2 + h \cdot 2 \cdot 3
|
15,387 |
g \cdot \dfrac{g_1}{g} \cdot n = g \cdot n/g \cdot g_1
|
44,245 |
\frac{2}{z + 2\cdot \left(-1\right) + 3\cdot \left(-1\right)} + 1 = \frac{\left(-1\right) + z + 2\cdot \left(-1\right)}{3\cdot (-1) + z + 2\cdot (-1)}
|
19,178 |
\frac100^2 = 0^{2 + (-1)}
|
3,572 |
\delta = 0,v \neq 0 \Rightarrow \frac{\delta^2 v}{\delta^4 + v^2} = 0
|
23,018 |
\cos(z_1 + z_2) = \cos(z_1) \cos(z_2) - \sin(z_1) \sin(z_2)
|
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