id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
-1,602 |
\frac34 \cdot \pi = -13/12 \cdot \pi + \pi \cdot 11/6
|
23,794 |
(-1)^{2 + m} = (-1)^m
|
18,390 |
h_1^2 = c \cdot h_2 + h_2^2 rightarrow -h_2^2 + h_1^2 = c \cdot h_2
|
17,004 |
\sqrt{\frac{7950}{30} - 16^2} = 3
|
14,981 |
8 - \dfrac89 = \dfrac{64}{9}
|
54,578 |
|x| = |x - y + y| \leq |x - y| + |y|
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54,998 |
2 + 6 + 120 = 128 = 2^7
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14,671 |
\left(2\cdot t^2\right)^2 = 2^2\cdot (t^2)^2 = 4\cdot t^4
|
-20,908 |
\frac{8\cdot s + 3}{3 + s\cdot 8}\cdot \left(-\frac51\right) = \frac{-40\cdot s + 15\cdot (-1)}{s\cdot 8 + 3}
|
-23,169 |
-1 = -\dfrac13*3
|
-20,448 |
\frac{4}{4} \cdot \frac{3 \cdot \left(-1\right) + y}{y + 3} = \frac{1}{y \cdot 4 + 12} \cdot (y \cdot 4 + 12 \cdot (-1))
|
27,338 |
\{E_1, E_2\} \implies E_2 = E_1 \cup E_2 \setminus E_1
|
23,622 |
(A*X)^2 = A*X*A*X = X*A*A*X = X*A^2*X
|
20,960 |
2 + 7578/24541 = \frac{56660}{24541}
|
-29,592 |
\frac{\text{d}}{\text{d}y} (2 \cdot y^2) = 2 \cdot \frac{\text{d}}{\text{d}y} y^2 = 2 \cdot 2 \cdot y^1 = 4 \cdot y
|
9,830 |
(P(Y) + P(B))^2 = P\left(Y\right)^2 + 2 P(Y) P\left(B\right) + P\left(B\right) P\left(B\right) = 1 \implies P(B)^2 + P(Y)^2 = 0.9
|
15,431 |
a^2 + b^2 = (b + a)^2 - 2ba
|
30,208 |
\left(p^2 \cdot p\right)^3 = (p^5)^5 = \left(p^7\right)^7 = p
|
27,350 |
a f = a f
|
31,426 |
5 + 2 \cdot 6^{1/2} = (2^{1/2} + 3^{1/2}) \cdot (2^{1/2} + 3^{1/2})
|
24,279 |
(c^3)^\beta + \beta = (c^3)^\beta + \beta^3 - \beta^3 + \beta = (c^\beta + \beta) \cdot ((c^2)^\beta - \beta \cdot c^\beta + \beta^2) - \beta^3 + \beta
|
14,518 |
(u - v \sqrt{V}) (u + v \sqrt{V}) = -v^2 V + u^2
|
11,784 |
1 - \dfrac12 - 1/5 = \frac{3}{10}
|
4,023 |
a^n = a^{n + 0} = a^n a^0
|
13,806 |
( r, z) \cdot ( t', \beta) := t' \cdot (-r) + \beta \cdot z
|
30,259 |
B = \sqrt{A*B} \Rightarrow B = 0\text{ or }B = A
|
21,777 |
x*(n + 1)*4 = x*(4 + 4n)
|
15,723 |
{l + k + \left(-1\right) \choose k + (-1)} = {(-1) + l + k \choose l}
|
26,517 |
(\left(-1\right) (-1) + 7)/2 = 4
|
1,398 |
\frac{d_2^2}{a\cdot d_2 + a\cdot d_1}\cdot d_1^2 = \frac{a^2\cdot d_1 \cdot d_1}{a\cdot d_2 + d_2\cdot d_1} = \tfrac{a^2\cdot d_2^2}{d_2\cdot d_1 + a\cdot d_1}
|
5,826 |
(a + x)/2 = a + \dfrac{1}{2} \left(x - a\right) = x - (x - a)/2
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35,942 |
A \cdot A + A = 3\cdot A - I = A \cdot A \cdot A + I
|
42,547 |
1/(\frac{1}{a}) = 1/(\dfrac1a) = 1/a\cdot a/(1/a) = \frac{1}{1/a\cdot a}\cdot a = a = a
|
2,952 |
\frac{2 x y}{x + y} = \frac{2}{\frac1x + \dfrac{1}{y}} \leq (x + y)/2
|
3,471 |
h_1/(g_1) + \frac{h_2}{g_2} = \frac{1}{g_2 g_1} \left(g_2 h_1 + h_2 g_1\right)
|
394 |
0*\ldots*2*π = 0
|
2,655 |
a^4 + h^4 = \left(a + h\right)^4 - 4 \cdot h \cdot a^3 - 6 \cdot h^2 \cdot a^2 - h^3 \cdot a \cdot 4
|
-9,473 |
2\cdot t + t\cdot 2\cdot 2\cdot t = 4\cdot t^2 + 2\cdot t
|
40,310 |
x^q = x^{q + (-1)}*x = 2^{(q + (-1))/2}*x
|
9,765 |
g^Q g^m = g^{Q + m}
|
-5,940 |
\frac{1}{2\cdot d + 4}\cdot 5 = \frac{1}{2\cdot \left(2 + d\right)}\cdot 5
|
16,245 |
d + b = 4\Longrightarrow d - b = 4
|
11,020 |
0 = x^2 + z^2 - x\cdot 2 - 2 b z + 8 (-1) \implies (\sqrt{b b + 9})^2 = (z - b)^2 + (x + (-1)) (x + (-1))
|
3,711 |
z + 3\cdot \left(-1\right) = \frac{1}{2}\cdot (z\cdot 2 + 6\cdot (-1))
|
4,522 |
\frac{9}{29} \cdot \tfrac{1}{30} \cdot 30 = \frac{1}{29} \cdot 9
|
27,337 |
(p^2 - p)\cdot (\left(-1\right) + p^2) = (p + 1)\cdot ((-1) + p)^2\cdot p
|
-19,438 |
8/3\cdot \frac29 = \frac{8\cdot 2}{3\cdot 9} = 16/27
|
10,102 |
\sin(\frac{\pi}{2} - t) = \cos{t}
|
36,002 |
e^{D + B} = e^D \cdot e^B = e^B \cdot e^D
|
7,255 |
1 + z^4 + z^2 = (z \cdot z + z + 1) \cdot (z^2 - z + 1)
|
30,606 |
z^2 + z + 8 = z^2 - 9 \cdot z + 8 = (z + (-1)) \cdot (z + 8 \cdot (-1)) = (z + 9) \cdot \left(z + 2\right)
|
-11,494 |
16 - 24 \cdot i = -24 \cdot i - 4 + 20
|
23,282 |
\tfrac{1}{H_1 \cdot H_2} \cdot \left(C \cdot H_1 + A \cdot H_2\right) = \frac{1}{H_2} \cdot C + A/(H_1)
|
8,319 |
x^2\cdot 2 = (x\cdot \sqrt{2})^2
|
28,426 |
((-5)^2)^{\tfrac{1}{2}} = |-5| = 5
|
-26,464 |
\phi \cdot \phi + 4^2 - \phi\cdot 4\cdot 2 = (4 - \phi)^2
|
12,962 |
135^2 = 3^2\cdot 45^2 = 9\cdot 2025 < 9\cdot 2040
|
22,446 |
c^2 \cdot d^2 = \left(d \cdot c\right) \cdot \left(d \cdot c\right)
|
-3,858 |
\frac{x^5}{x^5}\cdot 63/54 = \frac{x^5\cdot 63}{54\cdot x^5}\cdot 1
|
20,358 |
\tfrac{1}{b \cdot b} \cdot \tfrac{1}{b \cdot b} + (b^2)^2 = b^4 + \dfrac{1}{b^4}
|
6,546 |
-397\cdot 1027776565^2 + 20478302982^2 = -1
|
27,172 |
\left(\sin(b + a) + \sin(a - b)\right)/2 = \sin(a)\cdot \cos(b)
|
10,088 |
a^3 - c^3 = (c^2 + a \times a + a\times c)\times (a - c)
|
34,256 |
1/3 = \dfrac{3}{9}
|
9,639 |
(a*x)^2 = a^2*x^2
|
19,414 |
(g + f)^2 - fg \cdot 2 = f^2 + g^2
|
15,401 |
-h^3 + z^3 = (z - h) (h^2 + z^2 + hz)
|
10,694 |
z^2 = z^2 + (-1) + 1 = \left(z + 1\right)*\left(z + (-1)\right) + 1
|
-26,624 |
s^2 \cdot 81 - 144 \cdot s \cdot t + t \cdot t \cdot 64 = \left(9 \cdot s - 8 \cdot t\right) \cdot \left(9 \cdot s - 8 \cdot t\right)
|
18,575 |
\frac19 + 1/8 = \dfrac{17}{72}
|
22,857 |
\left(m + 1\right)^2 - 31\times m + 257 = m^2 + 2\times m + 1 - 31\times m + 257 = m^2 - 29\times m + 258
|
21,360 |
(z_2 \cdot z_2 + z_1^2)^2 = (z_2^2 - z_1^2)^2 + (2z_2 z_1) \cdot (2z_2 z_1) = 5^2 + 12 \cdot 12 = 13^2\Longrightarrow z_2^2 + z_1^2 = 13
|
443 |
a_{n + 1} = \left(1 + a_n\right)*(n + 1) \Rightarrow 1 + n = \frac{a_{1 + n}}{a_n + 1}
|
35,814 |
\theta\times n\times \dfrac{1}{(n + \left(-1\right))!}\times (n + 2\times (-1))! = \frac{n}{n + (-1)}\times \theta
|
-9,973 |
0.01 (-4) = -\frac{4}{100} = -0.04
|
28,832 |
3/8 + \dfrac28 = \frac{5}{8}
|
-19,795 |
125\% = \dfrac{125}{100} = 1.25
|
32,357 |
10^l = (2 \cdot 5)^l
|
-2,766 |
\sqrt{7} + \sqrt{9}\cdot \sqrt{7} = \sqrt{7} + \sqrt{7}\cdot 3
|
24,682 |
r*(r + 1) = r^2 + r
|
12,212 |
\frac{x}{x} \cdot s = s \cdot x/x
|
-6,388 |
\frac{20}{(6 + p)*(p + 4*(-1))*4} = \dfrac{5}{(4*(-1) + p)*(p + 6)}*\dfrac{4}{4}
|
46,291 |
105 = 53 + 52
|
31,915 |
1 + x^2 - 2\times x = (x + (-1))^2 rightarrow x^2 - 2\times x = \left(x + (-1)\right)^2 + (-1)
|
-20,512 |
\dfrac{1}{10 p + 60 (-1)}(10 + 10 p) = \dfrac{10}{10} \frac{1 + p}{6(-1) + p}
|
25,348 |
\int_{-2}^{-1} (5(-1) + y^2 c)\,\mathrm{d}y = 0 rightarrow \int\limits_1^2 (cy^2 + 5(-1))\,\mathrm{d}y = 0
|
19,983 |
2/(1/2\cdot x) = 4/x
|
3,332 |
\frac{x^2}{x + 9 \cdot (-1)} = x + \dfrac{81}{x + 9 \cdot (-1)} + 9
|
46,839 |
(\sqrt{z + 7} - \sqrt{14}) \cdot (\sqrt{z + 7} + \sqrt{14}) = z + 7 + 14 \cdot (-1) = z + 7 \cdot (-1)
|
31,552 |
x^3+1=x^3-1=(x-1)(x^2+x+1)=(x+1)(x^2+x+1)
|
18,305 |
(2^{20})^{10} = 2^{20}\cdot 2^{20}\cdot \ldots\cdot 2^{20}
|
423 |
(-1) + \frac{1}{Y - b} \cdot \left(t - b\right) = \frac{t - Y}{-b + Y}
|
13,531 |
\dfrac{13!}{7!} - \frac{12!}{7!} = 1140480
|
6,129 |
1/3 + 1/(4\cdot 3) = -1/(3\cdot 4) + \frac{1}{2}
|
32,298 |
-(x - e) = e - x
|
37,088 |
\frac{1}{2} \cdot (1 + 5^{1/2}) = \frac{1}{2} \cdot 5^{1/2} + \dfrac12
|
12,886 |
\dfrac{1}{9}\cdot 2 + \dfrac{5}{18} + 5/18 = 7/9
|
8,820 |
g \times e \times f = g \times e \times f
|
-5,688 |
\frac{1}{(\left(-1\right) + \xi) (\xi + 2)}4 = \dfrac{4}{\xi^2 + \xi + 2\left(-1\right)}
|
10,285 |
D_1^2 \times D_2^3 = D_1 \times D_1 \times D_2^3
|
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