id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
20,697 |
\arccos(\cos(-\arccos{p} + π)) = \arccos{-p} \implies \arccos{-p} = -\arccos{p} + π
|
4,147 |
0 = 1 + 2 + 3(-1)
|
-11,546 |
(4 + 5) + (-1i) = 9-i
|
-18,303 |
\frac{x \cdot \left(8 + x\right)}{(x + 8) (x + 5)} = \frac{x^2 + x \cdot 8}{40 + x^2 + 13 x}
|
15,813 |
x^j = x^j
|
35,713 |
z^2 + 1 = z^2 + 1
|
8,032 |
3 \cdot 2^m - 2 \cdot 2^{m + (-1)} = 3 \cdot 2^m - 2^m = 2^m \cdot (3 + \left(-1\right)) = 2^m \cdot 2 = 2^{m + 1}
|
1,479 |
12 z = y^2 \implies 3(z + 0(-1))*4 = (y + 0\left(-1\right))^2
|
2,145 |
(c + (-1))*(c + 1) = (c + (-1))*c + c + (-1) = c^2 - c + c + (-1) = c * c + (-1)
|
20,327 |
E = 1/\left(\tfrac{1}{E}\right)
|
10,983 |
k/6 = \dfrac{\left(\frac{k}{2}\right)^2}{k + k/2}
|
-3,032 |
\left((-1) + 4\right)\cdot \sqrt{6} = \sqrt{6}\cdot 3
|
-24,692 |
27 \cdot i + 7 = -11 + i \cdot 23 + 18 + 4 \cdot i
|
35,035 |
|-t\cdot 2 + 16| = 5\Longrightarrow t = 5.5,10.5
|
-5,657 |
\frac{2}{(k + 4\cdot (-1))\cdot \left(k + 8\cdot (-1)\right)} = \frac{2}{32 + k^2 - k\cdot 12}
|
14,350 |
|u_{l + 1} - u_l| = 1 \gt \frac{u_l u_{l + 1}}{(l + 1) \cdot (l + 1)}
|
-3,348 |
2^{\frac{1}{2}}\cdot 4 + 2^{1 / 2} = 2^{\frac{1}{2}} + 2^{\dfrac{1}{2}}\cdot 16^{1 / 2}
|
-13,086 |
2 \left(-2\right) = 2 (-2)/(1) = -4/1
|
19,294 |
2^{\frac{1}{3}} + \left(-1\right) = (x^{1/3} + y^{1/3} + z^{\frac13})^3 = x + y + z + \ldots
|
32,114 |
-b + a\cdot (b + \left(-1\right)) = -b + b\cdot a - a
|
-491 |
-\pi\cdot 8 + \frac{55}{6}\cdot \pi = \frac76\cdot \pi
|
23,613 |
\binom{m}{m} = \binom{m + (-1)}{m} + \binom{(-1) + m}{(-1) + m}
|
-25,237 |
d/dx \sqrt{x^5} = 5/2 x
|
31,816 |
\frac{1}{4}\cdot 0 + 3\cdot \frac14/32 = \frac{3}{128}
|
3,378 |
-a \cdot (-d) = d \cdot a
|
-19,674 |
24/7 = \frac{24}{7} \cdot 1
|
12,806 |
\dfrac{1}{(k! + (-1))!}(k!)! = k!/1!
|
18,999 |
\tfrac{1}{a^c} = a^{-c}
|
33,509 |
2^q = 4/4 = 1 \Rightarrow 0 = q
|
-20,103 |
\frac{63}{21\times (-1) + 7\times D} = \frac{1}{D + 3\times (-1)}\times 9\times \frac{1}{7}\times 7
|
46,310 |
\left\lfloor{k/2}\right\rfloor + \left\lceil{k/2}\right\rceil = \dfrac12(k + \left(-1\right)) + \dfrac{1}{2}(k + 1) = k
|
9,462 |
p^{\alpha} + \left(-1\right) = (p + \left(-1\right)) \cdot (p^{\alpha + (-1)} + p^{2 \cdot (-1) + \alpha} + \dots + p + 1)
|
16,495 |
1 + 5 + 5 \cdot 5 + \dotsm + 5^{x + \left(-1\right)} = \frac{1}{5 + (-1)} \cdot (5^x + (-1)) = \dfrac14 \cdot (5^x + (-1))
|
8,843 |
3 \cdot \pi/2 \lt t < 2 \cdot \pi,\sin{t} < 0\Longrightarrow \sin{t} = -(1 - \cos^2{t})^{1 / 2}
|
-29,493 |
-5 + \dfrac{63}{9} = -5 + 7 = -5 + 7 = 2
|
8,692 |
\dfrac{-q^{1 + n} + 1}{1 - q} = 1 + q + q^2 + \dots + q^n
|
3,463 |
\frac16 \cdot (2 \cdot z + 3) = \frac{z}{3} + 1/2
|
-9,331 |
3\cdot 3\cdot 7 + a\cdot 3\cdot 3\cdot 7 = 63 + 63\cdot a
|
11,695 |
A \cdot (r + s) = A \cdot r + A \cdot s
|
-20,980 |
\frac{1}{p*10 + 100}*(-p*9 + 90*(-1)) = -\frac{9}{10}*\frac{p + 10}{p + 10}
|
2,531 |
A' \cdot G \cdot x + x \cdot C = x \cdot (C + A' \cdot G)
|
1,399 |
\frac{1}{xZ \cdot 1.1}(-xB + Zx \cdot 1.1) = 1 - \frac{xB}{xZ \cdot 1.1}
|
21,681 |
\dfrac{1}{\mathbb{E}\left(A_1 \cdot A_1\right) + \mathbb{E}\left(A_2^2\right)} \cdot \mathbb{E}\left(A_1^2\right) = \mathbb{E}\left(\frac{A_1^2}{A_1^2 + A_2^2}\right)
|
28,613 |
Z = Z + 0
|
-7,079 |
\dfrac29*3/8 = 1/12
|
11,807 |
h^2 + f \cdot f + h^2\cdot f^2 = (h - f) \cdot (h - f) + 2\cdot h\cdot f + h^2\cdot f^2 = 1 + 2\cdot h\cdot f + h \cdot h\cdot f^2 = (1 + h\cdot f)^2
|
30,006 |
3^{2 r + 1} + 3^{2 r} = \left(3^r\cdot 2\right)^2
|
-7,528 |
(-x + h)\times (x + h) = -x \times x + h^2
|
-6,299 |
\frac{1}{648 \cdot (-1) + x^2 \cdot 12 - 36 \cdot x} \cdot \left(3 \cdot x + 27 \cdot (-1) + 12 \cdot x + 72 + 12 \cdot (-1)\right) = \frac{x \cdot 15 + 33}{12 \cdot x^2 - x \cdot 36 + 648 \cdot (-1)}
|
31,083 |
-m = m \Rightarrow m = 0
|
-12,758 |
7 + 8 + 7 = 22
|
16,683 |
36 = x \cdot x + z \cdot z \Rightarrow z = \sqrt{36 - x^2}
|
15,769 |
(x + 2) \cdot (x + x^2)/6 = {2 + x \choose 3}
|
4,170 |
\sin{3x} = \sin(x + 2x) = \sin{x} \cos{2x} + \cos{x} \sin{2x}
|
23,516 |
g^q \cdot g \cdot g^q \cdot g \cdot g^q \cdot g = g^q \cdot g \cdot g^q \cdot g \cdot g \cdot g^q
|
31,924 |
-736 = 1155 \cdot (-1) + 419
|
-27,686 |
d/dx (18 \sin(x)) = \cos(x) \cdot 18
|
-15,597 |
\frac{\frac1r \cdot a}{\dfrac{1}{\frac{1}{a^6} \cdot \frac{1}{r^6}}} = \frac{\dfrac{1}{r} \cdot a}{a^6 \cdot r^6} \cdot 1
|
-10,580 |
\dfrac{20}{20}\cdot \left(-\frac{3}{15\cdot a \cdot a \cdot a}\right) = -\dfrac{1}{a^3\cdot 300}\cdot 60
|
22,387 |
2520 = {6 \choose 1} \cdot {9 \choose 3} \cdot {5 \choose 1}
|
-20,669 |
-\frac{1}{10}\cdot \frac{t\cdot 3 + 4}{4 + 3\cdot t} = \frac{1}{40 + 30\cdot t}\cdot (4\cdot \left(-1\right) - 3\cdot t)
|
7,502 |
8/9 = 1/27 \cdot 8/(1/3)
|
24,130 |
\left((-1) + x\right) (2(-1) + x) (x + 3\left(-1\right)) = 6\left(-1\right) + x^3 - 6x \cdot x + x \cdot 11
|
25,068 |
1/2 = 0 + \frac14 + 1/8 + \ldots
|
-20,971 |
\frac{20\cdot (-1) + 35\cdot t}{-t\cdot 42 + 24} = -\dfrac{5}{6}\cdot \frac{1}{4 - 7\cdot t}\cdot (4 - t\cdot 7)
|
13,837 |
\frac{h!}{(h + 2 \cdot (-1))!} = h \cdot ((-1) + h)
|
24,219 |
\zeta f \cdot 2 = \zeta f \cdot 2
|
6,186 |
\frac{a^{3/2}}{a^{-1}} = a^{3/2}\cdot a = a^{5/2}
|
5,211 |
\frac1x \cdot \left(1 + x\right) = 1/x + 1
|
-18,186 |
4 = 48\cdot \left(-1\right) + 52
|
9,540 |
(3)\;\lim_{n\to\infty}a_n=a>0\implies\lim_{n\to\infty}\sqrt[n]{a_n}=1
|
-17,425 |
44 + 38\cdot \left(-1\right) = 6
|
7,702 |
(z_1 - z_2 + 3) \cdot (z_1 - z_2 + 3 \cdot (-1)) = \left(z_1 - z_2 + 3 \cdot \left(-1\right)\right) \cdot (z_1 - z_2 + 3)
|
10,612 |
\left(\beta + r\right)*\beta = \beta*\beta + r*\beta = \beta*\beta + r = \beta + r
|
23,848 |
(c^4 - c^2*6 + 1)^2 = 1 + c^8 - 12*c^6 + c^4*38 - 12*c^2
|
20,398 |
-\int\limits_0^1 \ldots\,dx = \int_1^0 \ldots\,dx
|
27,812 |
10 \cdot 4 \cdot \left(-27\right) = -1080
|
-6,004 |
\dfrac{a*4}{(a + 5*(-1))*(1 + a)} = \frac{a*4}{a^2 - 4*a + 5*\left(-1\right)}
|
18,021 |
(\left(-2\cdot x + \left(-1\right)\right)!)! = \dfrac{1}{((2\cdot x + \left(-1\right))!)!}\cdot (-1)^x = (-1)^x\cdot 2^x\cdot x!/(2\cdot x)!
|
18,744 |
(-\frac12 + x^2) \cdot (-\frac12 + x^2) + \dfrac{3}{4} = 1 + x^4 - x^2
|
-26,420 |
x^{\xi + n} = x^n\cdot x^\xi
|
49,806 |
\sqrt{1/(-1)} = \sqrt{-1} = i
|
13,942 |
\frac{9!}{3!\times 3!\times 3!} = 1680
|
17,102 |
H^2 - H \cdot g \cdot 2 + g^2 = (H - g)^2
|
-1,340 |
\dfrac{5}{\frac19 \cdot (-8)} \cdot 1/4 = 5/4 \cdot \left(-9/8\right)
|
-8,325 |
-\frac{1}{7}\cdot 42 = -6
|
23,397 |
z = \sin(y) \Rightarrow \sin^2\left(y\right) = z^2
|
45,548 |
\frac{500}{2} = 250
|
1,743 |
x^2\cdot \alpha^2 = \alpha^2\cdot x^2 \Rightarrow \alpha\cdot x = x\cdot \alpha
|
3,748 |
x \cdot x a + b = z^2 \Rightarrow b = z \cdot z - ax^2
|
17,989 |
1 = \frac{1}{4} + 5/8 + 2 \cdot (-1) + c_2 \Rightarrow c_2 = \frac18
|
1,483 |
x + z * z = z * z + x
|
18,961 |
(f_1 + 11)\cdot (f_2 + 11) - f_1\cdot f_2 = f_1\cdot f_2 + 11\cdot f_1 + 11\cdot f_2 + 11\cdot 11 - f_1\cdot f_2 = 11\cdot f_1 + 11\cdot f_2 + 121
|
-1,240 |
\frac{5}{(-4)*\frac19}*1/8 = -9/4*\frac{5}{8}
|
-7,629 |
\frac{1}{18}\cdot (9 + 63\cdot i + 9\cdot i + 63\cdot (-1)) = \dfrac{1}{18}\cdot (-54 + 72\cdot i) = -3 + 4\cdot i
|
15,257 |
\tfrac{1}{15} = \frac{1}{3 \cdot 5} = 19 \cdot 16 = 16 \cdot 3 \cdot 6 + 16 = 6 + 16 = 22
|
20,127 |
-1 = -2*5 \left(-1\right) + 11 (-1)
|
13,739 |
(1 + y)^D \cdot (1 + y)^r = \left(1 + y\right)^{r + D}
|
1,219 |
h^2 + a \cdot h \cdot 2 + a \cdot a = \left(h + a\right)^2
|
14,352 |
x*2 = 2300 \Rightarrow 1150 = x
|
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