id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
-11,175 |
(z + 8(-1))^2 + f = (z + 8(-1)) (z + 8(-1)) + f = z^2 - 16 z + 64 + f
|
23,136 |
\left|{A + G}\right| = 0 \lt \left|{A}\right| + \left|{G}\right|
|
24,497 |
\cos(\operatorname{atan}(x)) = \dfrac{1}{(1 + x^2)^{1/2}}
|
26,671 |
5/(9*8) + 1/(6*3) = \frac18
|
7,902 |
-z^3 + x^2 \cdot x = (x \cdot x + x \cdot z + z^2) \cdot \left(-z + x\right)
|
51,248 |
3 + 1 + 2 = 3 + 2 + 1
|
33,256 |
72*\left(-1\right) + 145 = 73
|
21,399 |
\dfrac{4}{p + 2*(-1)} + 1 = \frac{1}{p + 2*\left(-1\right)}*(2 + p)
|
23,709 |
-\left(-6\right) + 2*\left(-2\right) = 2
|
-29,355 |
\left(2 + 7x\right) (2 - 7x) = 2^2 - (7x)^2 = 4 - 49 x^2
|
12,852 |
d \cdot d + x^2 + d\cdot x\cdot 2 = (x + d) \cdot (x + d)
|
4,856 |
-x^2 + 1 = \left(1 - x\right)*(1 + x)
|
33,233 |
2000 = \frac12 \cdot 800 + \frac{3200}{2}
|
27,414 |
\ln(z + 1) = z - \dfrac12*z^2 + z^2 * z/3 - \frac14*z^4 + \dots
|
17,251 |
h = (w/h + h/w) \cdot x = \frac{1}{w \cdot h} \cdot (w^2 + h^2) \cdot x
|
15,378 |
(-\cos(\alpha + \beta) + \cos(\alpha - \beta))/2 = \sin\left(\beta\right) \cdot \sin(\alpha)
|
15,655 |
\tfrac{1}{C_x\cdot C}\cdot (C - C_x) = 1/\left(C_x\right) - 1/C
|
2,647 |
x^{N + 1} = x \cdot x^N
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14,003 |
1 > |x/2| \implies |x| < 2
|
18,142 |
\cos^2(x) = \frac12\cdot (1 + \cos\left(x\cdot 2\right))
|
1,326 |
(n + 1)! = (n + 1) \cdot n! \lt (n + 1) \cdot n^n = n^{n + 1} + n^n \lt n^{n + 1} + n^{n + 1} = 2 \cdot n^{n + 1} < (n + 1)^{n + 1}
|
-19,673 |
18/9 = \dfrac{2*9}{9}
|
19,046 |
\frac{3}{3 + 2}*\frac{3}{3 + 2} = \frac{1}{25}*9 = 0.36
|
10,353 |
1 + x^2 \leq x^2*2 \Rightarrow \frac{1}{1 + x^2} \geq \frac{1}{2x^2}
|
21,498 |
33 \cdot 3^m = -3 \cdot 3^m + 3^m \cdot 9 + 3^m \cdot 27
|
-20,731 |
10/7*\frac{1}{5*(-1) + r}*\left(5*(-1) + r\right) = \frac{10*r + 50*(-1)}{7*r + 35*(-1)}
|
-6,017 |
\tfrac{1}{5 \cdot q + 15} \cdot 4 = \frac{4}{(3 + q) \cdot 5}
|
757 |
\left(\sqrt{j}\right)^2 = j = 0 + j = (c + x \cdot j) \cdot (c + x \cdot j) = c^2 + 2 \cdot c \cdot x \cdot j + x^2 \cdot j^2 = c^2 - x \cdot x + 2 \cdot c \cdot x \cdot j
|
13,291 |
\sum_{m=1}^l \left(1 + m\right) (m + (-1)) = \sum_{m=1}^l ((-1) + m) (1 + m)
|
15,845 |
c*d = \frac{1}{c*d} = 1/(d*c) = d*c
|
4,567 |
(2^{20} + (-1)) \cdot 3 = -3 + 2^{20} \cdot 3
|
13,655 |
7 = y + (-y^2 + 9\cdot y + 1)/7 = (-y^2 + 16\cdot y + 1)/7
|
7,758 |
2*x + (-1) < x + 1 \implies x \lt 2
|
3,958 |
\frac{2}{24} = \frac{1}{15 + 2 + 7} \cdot 2
|
41,284 |
\frac{1}{5} 3 = \dfrac15 3
|
13,607 |
\frac{1}{\cos(h)}*\sin(h) = \tan(h)
|
-14,069 |
6 + (9 \times 4) = 6 + (36) = 6 + 36 = 42
|
-3,470 |
\dfrac{5\cdot 9}{20\cdot 5} = 45/100
|
-23,084 |
2(-\frac23) = -4/3
|
5,795 |
n\cdot 2^{n + (-1)} = (\sum_{k=0}^n \binom{n}{k}) k = (\sum_{k=1}^n \binom{n}{k}) k
|
-6,483 |
\tfrac{2\cdot q}{(\left(-1\right) + q)\cdot \left(7\cdot (-1) + q\right)} = \dfrac{2\cdot q}{7 + q^2 - q\cdot 8}
|
-29,363 |
(y + 4) (y + 6) = y^2 + 6 y + 4 y + 24 = y^2 + 10 y + 24
|
18,811 |
\frac1x \times x^3 = r \implies r = x^2
|
14,336 |
-x^2 + d^2 = (-x + d) \cdot (x + d)
|
24,703 |
y^{12} + (-1) = ((-1) + y^6)*(y^6 + 1)
|
-20,056 |
\frac19\cdot 9\cdot \frac{1}{i\cdot (-9)}\cdot (1 + 5\cdot i) = \frac{9 + 45\cdot i}{i\cdot (-81)}
|
4,151 |
3 \cdot h \geq 0 \implies h = |h|
|
8,985 |
\frac{1}{R \cdot U} = \frac{1}{R \cdot U}
|
-22,224 |
x^2 + 9x + 20 = (x + 5) (4 + x)
|
-20,268 |
\frac{8\cdot q}{q\cdot 72} = 8\cdot q\cdot 1/(8\cdot q)/9
|
30,309 |
2^{n + (-1)}\cdot (2^n + 1 + 2\cdot (-1)) + (-1) = 2^{n + (-1)}\cdot (2^n + 1) - 2^n + \left(-1\right) = (2^n + 1)\cdot (2^{n + (-1)} + \left(-1\right))
|
47,507 |
\frac{\mathrm{d}z}{\mathrm{d}x} = \frac{-h_x + h/g \cdot g_x}{h_z - \frac{h}{g} \cdot g_z} = \frac{-g \cdot h_x + h \cdot g_x}{g \cdot h_z - h \cdot g_z}
|
-3,178 |
-\sqrt{9} \sqrt{7} + \sqrt{16} \sqrt{7} = \sqrt{7}\cdot 4 - \sqrt{7}\cdot 3
|
11,321 |
\frac{1}{2 \cdot (-1) + 5^{1/2}} = \frac{\frac{1}{5^{1/2} + 2}}{5^{1/2} + 2 \cdot (-1)} \cdot (2 + 5^{1/2})
|
53,472 |
16 \cdot 16 = 256
|
4,433 |
\frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac13
|
-2,529 |
2\sqrt{5} = ((-1) + 3) \sqrt{5}
|
19,240 |
3/2 - \dfrac23 = 5(\frac{1}{2} - \dfrac{1}{3})
|
14,804 |
1/9 + \frac{4*\frac13}{18} = \frac{1}{27}*5
|
21,902 |
30^{2010} \cdot 67^{2011} = \left(30 \cdot 67\right)^{2010} \cdot 67
|
15,396 |
|A^\complement \cap E^\complement| = |D \backslash A \cup E| = |D| - |A| - |E| + |A \cap E|
|
-13,251 |
9 \times (1 + 2) = 9 \times 3 = 27
|
14,920 |
-x \cdot 2 = d/dx \left(-x^2\right)
|
6,888 |
2/3 = \dfrac{\frac{2}{9}}{\dfrac13} \cdot 1
|
12,495 |
e^{F + x} = e^F \times e^x = e^x \times e^F
|
32,407 |
0 = s^2 - x \cdot s - x \Rightarrow \dfrac{1}{2} \cdot (x \pm \sqrt{x^2 + 4 \cdot x}) = s
|
28,091 |
\sin(D + G) = \cos{D} \times \sin{G} + \cos{G} \times \sin{D}
|
2,103 |
x^5 + x + (-1) = x^5 + x^2 - x^2 + x + \left(-1\right) = (x^2 - x + 1) (x^3 + x^2 + (-1))
|
7,042 |
1/N = \frac{1}{N \cdot N}\cdot N
|
-6,730 |
\frac{1}{10}3 + 3/100 = \frac{30}{100} + \frac{3}{100}
|
23,967 |
1 + 2m - q + 1 = 2m - q
|
-2,014 |
13/12\cdot \pi - \pi/6 = \dfrac{11}{12}\cdot \pi
|
-1,950 |
5/3 \cdot \pi + \frac{17}{12} \cdot \pi = \pi \cdot 37/12
|
1,460 |
\frac{1}{n^2} \times (n + (-1)) \times (1 + n) = 1 - \dfrac{1}{n^2}
|
26,709 |
-(-z + m)^2 + (m + z)^2 = z \cdot m \cdot 4
|
-25,801 |
\frac{5*1/6}{3} = \frac{5}{18}
|
8,471 |
-(-y^2 + A \cdot A) \cdot \sqrt{-y^2 + A^2}/3 + Z = Z - (A^2 - y^2)^{\frac12 \cdot 3}/3
|
-6,695 |
3/100 + 5/10 = 3/100 + \frac{50}{100}
|
-12,751 |
\dfrac36 = \dfrac{1}{2}
|
16,840 |
q^2\cdot (\left(-1\right) + q)/2 = (-q^2 + q^3)/2
|
16,617 |
\cos^2(x) \cdot 2 = -2\sin^2(x) + 2
|
26,532 |
24/(\sqrt{6}) = \sqrt{6}/\left(\sqrt{6}\right)\cdot \frac{1}{\sqrt{6}}\cdot 24
|
-16,780 |
{-4k} = ({-4k} \times -2k) + ({-4k} \times 6) = (8k^{2}) + (-24k) = 8k^{2} - 24k
|
15,389 |
\tfrac{1}{f + d}*(f^2 - d^2) = f - d
|
-9,935 |
-0.88 = -\tfrac{22}{25}
|
29,323 |
14 = 27 + 13\cdot \left(-1\right)
|
-25,825 |
\frac{5\cdot 1/4}{7\cdot 1/5} = 5/4\cdot \tfrac17\cdot 5 = \frac{5\cdot 5}{4\cdot 7} = 25/28
|
10,955 |
1 - \frac{2}{15} = \dfrac{13}{15}
|
1,188 |
l \cdot l \cdot l + 3 \cdot (-1) = \frac{1}{2} \cdot l^3 + \dfrac12 \cdot l^2 \cdot l + 3 \cdot (-1) > \frac{1}{2} \cdot l^2 \cdot l
|
32,210 |
{n + 1 \choose n} + 1 = {n + 2 \choose 1 + n}
|
11,966 |
0 = s^2 - s \implies s = 0,1
|
12,452 |
z^2 = -y^2 + 2 - z \cdot z \Rightarrow 1 = z^2 + y^2/2
|
25,899 |
\dfrac16\cdot 312 = 52
|
34,924 |
\delta_{x_i} = \frac12\cdot \delta_{x_i}
|
751 |
(2^{(-1) + l} + (-1)) \cdot (1 + 2^l) = (-1) + {2^l \choose 2}
|
11,163 |
(1 + x) \cdot (x + (-1)) = (-1) + x \cdot x
|
48,313 |
{1 + 5 \choose 1} = {6 \choose 1}
|
30,668 |
0.6*(\left(-1\right)*0.4 + 1)*\left((-1)*0.2 + 1\right) = 0.288
|
326 |
-\tfrac{2}{4^{1/2}} = -\tfrac{1}{2}2 = -1
|
19,682 |
2*\dfrac{1}{(2*y)^2 + 1}/3 = d/dy (\operatorname{atan}\left(y*2\right)/3)
|
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