id
int64 -30,985
55.9k
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stringlengths 5
437k
|
|---|---|
26,804
|
\sqrt{\frac{20}{12}} = \sqrt{\frac53}
|
21,722
|
\dfrac{1}{10} + 1/15 = \frac{1}{30}\cdot 3 + 2/30 = \frac{1}{30}\cdot 5
|
4,369
|
\left((1 + \sqrt{13})/2\right)^3 = \sqrt{13}*2 + 5
|
33,608
|
35/12 = (6^2 + \left(-1\right))/12
|
1,624
|
(1 + y\cdot 2)\cdot (2\cdot y + 1)\cdot (1 + 4\cdot y)\cdot (1 + y) = 1 + 9\cdot y + 28\cdot y \cdot y + y \cdot y \cdot y\cdot 36 + 16\cdot y^4
|
21,941
|
\left(k + 1\right)^2 - (k + \left(-1\right))^2 = k^2 + 2\cdot k + 1 - k^2 + 2\cdot k + (-1) = 4\cdot k
|
-531
|
\pi \frac{2}{3} = 32/3 \pi - \pi \cdot 10
|
3,280
|
2\sqrt{3} + 4 = 3 + \sqrt{3} \cdot 2 + 1
|
35,258
|
\frac{a}{f + c} = \frac{f}{c + a} = \frac{1}{a + f}\cdot c
|
2,846
|
x \cdot x - x\cdot 2 + 1 = \left(x + (-1)\right)\cdot \left(\left(-1\right) + x\right)
|
20,026
|
\frac{BA}{A} = G/A rightarrow \frac1AG = B
|
28,315
|
(1 + \cos{2y})^2 = \left(1 + 2\cos^2{y} + \left(-1\right)\right) \cdot \left(1 + 2\cos^2{y} + \left(-1\right)\right) = 4\cos^4{y}
|
-21,004
|
\frac{3}{10}\cdot 9\cdot l/\left(9\cdot l\right) = 27\cdot l/(90\cdot l)
|
43,391
|
(x+y)+z= x+(y+z)
|
14,437
|
\sum_{F=1}^k (c\cdot a_F + b_F) = c\cdot \sum_{F=1}^k a_F + \sum_{F=1}^k b_F
|
29,667
|
{n + t \choose n} = {t + n \choose t}
|
39,098
|
5^5 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5
|
26,821
|
\frac12\times 15 = 7.5\times \dotsm
|
-3,344
|
2^{\frac{1}{2}} + (16\times 2)^{\dfrac{1}{2}} = 32^{1 / 2} + 2^{\tfrac{1}{2}}
|
-6,688
|
\frac{1}{100}60 + \frac{9}{100} = 9/100 + 6/10
|
-16,827
|
-6 = -6*5*p - -36 = -30*p + 36 = -30*p + 36
|
26,123
|
\binom{202}{2} = \frac{40602}{2}1 = 20301
|
28,160
|
\frac{1}{\sqrt{n}} = \sqrt{n}/n = \sqrt{n}/n
|
3,879
|
\frac{1}{A + x\cdot v^Y} = -\frac{x\cdot v^Y/A\cdot 1/A}{v^Y\cdot x/A + 1} + 1/A
|
10,248
|
\frac{1}{156} + 1/13 = 1/12
|
8,124
|
\tfrac{26}{21} = 1 + \frac{1}{21} \cdot 5 = 1 + \frac{1}{21 \cdot 1/5} = 1 + \frac{1}{4 + \dfrac{1}{5}}
|
33,783
|
n^2 + 3 = (n + 1) \cdot \left(n + (-1)\right) + 4
|
-27,624
|
-4 + 5 \cdot (-1) + 4 + 5 \cdot \left(-1\right) = -4 + 4 + 5 \cdot (-1) + 5 \cdot (-1) = 0 + 10 \cdot (-1) = -10
|
23,355
|
n^3 \leq 5 \cdot (-1) + 2 \cdot n^3 \implies 5 \leq n^3
|
15,199
|
-\frac{1}{5*\left(4/5 + (-1)\right)} = -1/(5*(-1/5)) = 1
|
15,201
|
-1/b = \dfrac{1}{b\cdot (-1)}
|
16,364
|
\frac11*0 + 2 = \frac21
|
21,919
|
5 + 3 \times m = 3 \times \left(1 + m\right) + 2
|
8,042
|
5 = -3 \cdot 10 + 5 \cdot 7
|
23,862
|
((\alpha^2 + 1)^{\frac{1}{2}} + t - \alpha) \cdot \left(t - \alpha - (1 + \alpha^2)^{\frac{1}{2}}\right) = t^2 - 2 \cdot \alpha \cdot t + \left(-1\right)
|
-7,162
|
\frac{3}{22} = \dfrac{3}{11}\cdot \frac{6}{12}
|
11,160
|
p^{n + 2\cdot \left(-1\right)}\cdot p^2 = p^{n + 2\cdot (-1) + 2} = p^n
|
-15,572
|
\frac{1}{z^3 a^3 z^4} = \frac{1}{a^3 z^3 z^4}
|
-2,593
|
\sqrt{175}+\sqrt{112}-\sqrt{63} = \sqrt{25 \cdot 7}+\sqrt{16 \cdot 7}-\sqrt{9 \cdot 7}
|
14,843
|
18 = -z_2\cdot 3 + z_3\cdot 5\Longrightarrow -z_2\cdot 3 = 18 - 5\cdot z_3
|
-12,777
|
\dfrac{12}{21} = \tfrac47
|
-14,135
|
\frac{1}{8 + 3}77 = 77/11 = \dfrac{1}{11}77 = 7
|
140
|
a_1\times a_2 + e\times a_2 = \left(a_1 + e\right)\times a_2
|
26,963
|
y\cdot c^b\cdot x\cdot c^a = c^a\cdot x\cdot c^b\cdot y
|
6,730
|
a^{b_1} a^{b_2} = a^{b_2 + b_1}
|
-1,329
|
\frac97*(-4/3) = \frac{9*1/7}{(-3)*1/4}
|
24,116
|
\dfrac{1}{2\left(x + (-1)\right)}(-x + 1) e^{(-1) + x} = -\tfrac{1}{2}e^{(-1) + x}
|
3,213
|
(a^2\cdot b \cdot b)^2 = ((b\cdot a)^2)^2 = a^4\cdot b^4
|
-26,509
|
(9 \cdot x)^2 + 9 \cdot x \cdot 10 \cdot 2 + 10^2 = (10 + 9 \cdot x)^2
|
19,140
|
\tfrac{1}{1 + n}*(n + 2) = 1 + \frac{1}{n + 1}
|
34,313
|
0 = 2 + z \implies -2 = z
|
26,539
|
(-a \cdot y_0 + z_0 \cdot b) \cdot (-a \cdot y_0 - z_0 \cdot b) = -b^2 \cdot z_0^2 + y_0 \cdot y_0 \cdot a^2
|
4,218
|
\frac{1}{2\cdot 3}\cdot \pi = \pi/6
|
9,502
|
(2 - \frac12\cdot 3) \cdot (2 - \frac12\cdot 3) = (-3/2 + 1)^2
|
26,173
|
\sum_{i=1}^\infty |-I_i| = \sum_{i=1}^\infty |I_i|
|
12,275
|
\frac1az^2 a = (a\frac{z}{a})^2
|
-5,889
|
\frac{3}{(9\cdot \left(-1\right) + z)\cdot 4} = \frac{1}{36\cdot (-1) + 4\cdot z}\cdot 3
|
13,382
|
-\int\limits_w^c 1\,\mathrm{d}z = \int\limits_c^w 1\,\mathrm{d}z
|
34,277
|
g_3 \cdot g_1 \cdot g_2 = g_1 \cdot g_2 \cdot g_3
|
37,392
|
(-x + 1) (1 + x + x^2 + x^3) = 1 - x^4
|
23
|
\|1/B\| \cdot \|\frac{1}{1/B}\| = \|1/B\| \cdot \|B\|
|
-3,726
|
\dfrac{r^5}{r^4} \cdot 40/5 = \frac{40 \cdot r^5}{5 \cdot r^4}
|
10,405
|
\dfrac{1}{10}*\left(61^2 + 9\right) = ((60 + 1) * (60 + 1) + 9)/10 = (3600 + 121 + 9)/10 = 373
|
50,998
|
\dfrac{1}{x + (-1)} \cdot (1 + x + x \cdot x + ... + x^n - n + 1) = \frac{1}{x + (-1)} \cdot (x + (-1) + x^2 + (-1) + ... + x^n + \left(-1\right)) = \dfrac{1}{x + \left(-1\right)} \cdot (x + \left(-1\right)) \cdot (1 + x + 1 + (x^2 + x + 1) \cdot ... + x^{n + (-1)} + x^{n + 2 \cdot (-1)} + ... + x + 1)
|
2,684
|
\frac{1}{28} \cdot 3 = 3 \cdot 1/7/4 + 4/7 \cdot 0
|
29,094
|
(y + (-1)) \cdot \left(y^2 + y + 2 \cdot \left(-1\right)\right) + 3 \cdot y + 3 \cdot (-1) = y^3 + \left(-1\right)
|
-7,647
|
3/(-1) - 3*i/(-1) = \frac{1}{-1}*(-i*3 + 3)
|
11,236
|
60 * 60 + 80 * 80 = 10000
|
27,627
|
648 = 2*9*\binom{9}{2}
|
-5,996
|
\dfrac{5}{3(r - 9)} \times \dfrac{r - 4}{r - 4} = \dfrac{5(r - 4)}{3(r - 9)(r - 4)}
|
4,389
|
2 \cdot g + h_1 = 71 \implies h_1 = 71 - 2 \cdot g = 71 - 2 \cdot (-4) = 71 + 8 = 79
|
15,958
|
{m + l \choose l} = {m + l \choose m}
|
12,075
|
1 - \omega + 2^n + (-1) = -\omega + 2^n
|
-23,348
|
4/9 \cdot \frac34 = 1/3
|
5,476
|
\sin\left(-h + a\right) + \sin(a + h) = 2\sin(a) \cos(h)
|
-18,333
|
\frac{(2(-1) + c) c}{(c + 2(-1)) (c + 10 (-1))} = \tfrac{c^2 - c*2}{c^2 - 12 c + 20}
|
16,444
|
1/3 + 1/5 + \frac{1}{6} = \frac{7}{10} \lt 3/4
|
535
|
Ax_0 = f \Rightarrow x_0 = f/A
|
2,145
|
(x + (-1)) \cdot (x + 1) = (x + (-1)) \cdot x + (x + (-1)) = x^2 - x + x + (-1) = x^2 + (-1)
|
3,448
|
e_i^t\cdot A\cdot e_j = \left(A^t\cdot e_i\right)^t\cdot e_j = (A\cdot e_i)^t\cdot e_j
|
25,238
|
\cot(-x + \frac{\pi}{4}) = \tan\left(x + \pi/4\right)
|
-26,465
|
64 - 16\cdot y + y^2 = 8 \cdot 8 - 2\cdot y\cdot 8 + y^2
|
-19,618
|
\frac{6 \cdot 1/7}{6 \cdot 1/5} = 6/7 \cdot \frac56
|
4,811
|
\left(\frac12 + \lambda\right)\cdot (-\frac{1}{2} + \lambda) = -1/4 + \lambda^2
|
20,008
|
\frac{1}{(-q + 1)^2}*(1 - \left(1 + n\right)*q^n + q^{1 + n}*n) = 1 + 2*q + 3*q^2 + \cdots + q^{(-1) + n}*n
|
19,257
|
\frac{d}{dy} (r*\sqrt{r}) = r/(2*\sqrt{r}) + \sqrt{r} = 3/2*\sqrt{r}
|
29,811
|
24 = 3! \cdot 2! \cdot 2!
|
-11,626
|
8 - 8*i = -i*8 + 0 + 8
|
27,658
|
x - X \cup Y = x \cap (X \cup Y)
|
15,564
|
\frac{\text{d}}{\text{d}z} \sin^{-1}(\frac1z) = -\frac{1}{z\cdot \sqrt{z \cdot z + (-1)}}
|
10,589
|
2\cdot x + (-1) = -\cos(2\cdot \sin^{-1}{x^{1/2}}) = 2\cdot \sin^2(\sin^{-1}{x^{1/2}}) + \left(-1\right) = 2\cdot x + (-1)
|
-30,812
|
x^2*10 + 30 = 10 (x^2 + 3)
|
26,029
|
\sinh(r) + \cosh(r) = e^r
|
34,443
|
g^2 - f^2 \cdot 4 = (g - 2 \cdot f) \cdot (2 \cdot f + g)
|
22,041
|
-2*x = -x - x
|
-5,577
|
\dfrac{4}{(8 + x) \cdot (x + 1)} = \frac{1}{x^2 + 9 \cdot x + 8} \cdot 4
|
34,353
|
h\cdot \int 1\,\mathrm{d}x := \int h\,\mathrm{d}x := \int h\,\mathrm{d}x
|
-19,622
|
\frac{\frac{1}{3} \cdot 8}{5 \cdot 1/3} = \frac83 \cdot \dfrac15 \cdot 3
|
7,943
|
-\zeta \gt 0 \Rightarrow \zeta < 0
|
-9,403
|
-3\times 2\times 2\times 2\times 3 + k\times 2\times 2\times 2\times 5 = 40\times k + 72\times (-1)
|
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