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25,231 | \lim_{\alpha \to 0} \frac{1}{\alpha^{1/3}} \cdot 2^{2/3} = \lim_{\alpha \to 0} \frac{2^{\frac{2}{3}}}{\alpha^{4/3}} \cdot \alpha |
-10,528 | \dfrac{36}{4*k + 12*(-1)} = \dfrac{9}{k + 3*(-1)}*4/4 |
-5,856 | \frac{4}{x^2 + x \times 5 + 4} = \frac{4}{(1 + x) \times \left(x + 4\right)} |
2,536 | \overline{C \cup X} = \overline{C} \cup \overline{X} = C \cup X |
21,843 | \dfrac{1}{2} + \frac13 + 1/6 = 1 |
-20,792 | \frac{k\cdot (-5)}{-k\cdot 6 + 2}\cdot 8/8 = \dfrac{(-1)\cdot 40\cdot k}{16 - 48\cdot k} |
12,034 | \cos(z) = \tfrac{-\tan^2\left(z/2\right) + 1}{1 + \tan^2(\frac{z}{2})} |
39,355 | b\times C = b\times C |
-7,626 | \dfrac{-1+13i}{2-i} = \dfrac{-1+13i}{2-i} \cdot \dfrac{{2+i}}{{2+i}} |
40,965 | 375 = 3\cdot 5 \cdot 5 \cdot 5 |
16,917 | l*p = l*p |
31,288 | (2^4)^{2^{2\cdot (-1) + t}} = 2^{2^t} |
20,050 | \cos\left(3 \cdot \left(\theta + \frac{\pi}{3}\right)\right) = \cos\left(2 \cdot \pi + 3 \cdot \theta\right) = \cos(3 \cdot \theta) |
28,181 | 20 + \dfrac{52}{6} = \frac16\cdot 172 |
27,802 | \omega^n + \overline{\omega}^n = \omega^n + \overline{\omega^n} = 2 \cdot \operatorname{re}{(\omega^n)} |
-26,510 | 128 - 32*z + 2*z * z = 2*(64 - 16*z + z^2) = 2*(8 - z)^2 |
9,776 | (x + y)\cdot (x + n\cdot y) = x^2 + n\cdot x\cdot y + x\cdot y + n\cdot y^2 \geq x \cdot x + (n + 1)\cdot x\cdot y |
28,803 | -s\cdot 41 + e\cdot 24 = -s + (-s\cdot 10 + 6\cdot e)\cdot 4 |
19,330 | \frac{z \cdot z \cdot 4 - z \cdot 5}{2 + 2 \cdot z^2 - 5 \cdot z} = 2 + \frac{5 \cdot z + 4 \cdot (-1)}{2 + z \cdot z \cdot 2 - z \cdot 5} |
2,501 | \frac{dy}{dz} = \frac{2 \cdot y + 2 \cdot z}{2 \cdot y - 2 \cdot z} = \dfrac{y + z}{y - z} |
-13,218 | \dfrac{1}{2/5 \cdot (-1/3)} \cdot 3 / 5 = \frac{3 \cdot 1/5}{2 \cdot (-1) \cdot \dfrac{1}{5 \cdot 3}} = \dfrac{3 \cdot \frac{1}{5}}{(-2) \cdot 1/15} = 3/5 \cdot (-15/2) = \dfrac{3 \cdot (-15)}{5 \cdot 2} = -\frac{45}{10} |
-11,639 | 2 + 5 + 9i = 7 + i\cdot 9 |
-22,993 | 21/27 = \dfrac{7}{9\cdot 3}\cdot 3 |
11,876 | \dfrac{\left(-1\right)*3 \pi}{4} = -2\pi + \dfrac{5\pi}{4} |
12,915 | \frac{\sin(2*m)}{\sin(m)*2} = \cos(m) |
16,341 | \frac{1}{\cos(y) + 1} = \frac{\text{d}}{\text{d}y} \left(\dfrac{\sin(y)}{1 + \cos(y)}\right) |
1,770 | x^2 + (z + \left(-1\right))\cdot \left(1 + z\right) = x^2 + z^2 + (-1) |
11,711 | E\left[-C\right] = -E\left[C\right] |
-6,261 | \frac{r}{r^2 + r \cdot 3 + 70 \cdot (-1)} = \frac{r}{(r + 10) \cdot (7 \cdot (-1) + r)} |
48,119 | 7 \times 43 = 301 |
38,267 | \frac{1}{(3 + 2 \times (-1))!} \times 3! = 6 |
19,551 | 1 = -k*2 + n \Rightarrow k = (n + \left(-1\right))/2 |
15,950 | z_1^4 + z_2^4 - 9\cdot z_1^2\cdot z_2^2 - 2\cdot z_1^2\cdot z_2^2 = z_1^4 + z_2^4 - 11\cdot z_1 \cdot z_1\cdot z_2^2 |
3,929 | 2 + 6*n = 2 + 3*n*2 |
3,272 | \dfrac{z^2}{z^4} = \dfrac{1}{z^2} |
-18,596 | \dfrac{1}{7} \cdot 3 = \dfrac37 |
31,166 | 0 \lt -(k \cdot k - k + 1) \cdot 64 + 64 \cdot k^2 \Rightarrow 1 \lt k |
-20,460 | \frac{2 - f*2}{2 - f*2}*\left(-\frac97\right) = \frac{18*f + 18*\left(-1\right)}{14 - 14*f} |
-11,556 | 3 + 0\cdot (-1) + 2\cdot i = i\cdot 2 + 3 |
39,820 | \sqrt{n * n - n + 1} - cn = \frac{(\sqrt{n * n - n + 1} - cn) (\sqrt{n^2 - n + 1} + cn)}{\sqrt{n * n - n + 1} + cn} = \frac{n * n - n + 1 - c^2 n^2}{\sqrt{n * n - n + 1} + cn} |
15,459 | \sin(y + \pi) = \sin\left(-y\right) |
28,979 | (-b + x) (x + b) = x x - b^2 |
-20,816 | \tfrac{1}{8 \cdot (-1) + q} \cdot (8 \cdot (-1) + q) \cdot \left(-5/7\right) = \frac{-q \cdot 5 + 40}{56 \cdot (-1) + q \cdot 7} |
23,721 | 1 + p^2 \cdot 64 + p \cdot 16 = (p + 1/8)^2 \cdot 64 |
-19,859 | 160\% = \dfrac{160}{100} = 1.6 |
-18,264 | \frac{1}{k^2 + 6k}(54 (-1) + k^2 - 3k) = \frac{1}{(6 + k) k}(k + 6) (k + 9\left(-1\right)) |
16,538 | \frac{1}{2} \times Y \times x^p \times x = x^p \times Y \times x/2 |
35,712 | -\frac{1}{10} + 1/30 + 1/15 = 0 |
17,214 | \frac{4^2}{12} = \dfrac{16}{12} = 4/3 |
-20,904 | \frac{x + 10\cdot \left(-1\right)}{1 - x\cdot 7}\cdot 9/9 = \frac{1}{9 - x\cdot 63}\cdot (90\cdot (-1) + x\cdot 9) |
20,157 | 1 + 2(-1) + 4 + 8(-1) + 16 ... = 1/3 |
5,833 | 2\cdot (1 + 1/4) = \dfrac{5}{2} |
-26,386 | \dfrac{z^n}{z^k} = z^{-k + n} |
-29,124 | \left(-1\right) (-2) + 3 = 5 |
13,467 | 2\cdot x \cdot x + 6\cdot x + 35 = 2\cdot (x^2 + 3\cdot x) + 35 = 2\cdot (x + \frac{1}{2}\cdot 3) \cdot (x + \frac{1}{2}\cdot 3) + \frac{61}{2} |
-18,707 | \left(-1\right)\cdot 0.0401 + 0.9332 = 0.8931 |
28,540 | {n + (-1) \choose n + (-1)} = {\left(-1\right) + n + (-1) + 1 \choose (-1) + n} |
15,108 | \sin(-\tfrac{\pi}{2}) = -1 |
37,106 | 11!*2 = 10!*11*2 |
17,497 | 1/2 = 1/100 + 49/99\cdot \frac{1}{100}\cdot 99 |
24,325 | \frac{1}{2^{10}}*144 = \tfrac{1}{2^{10}}*(6 + 1 + 10 + 36 + 56 + 35) |
28,723 | a^2 - x^2 = (a + x)\cdot (a - x) |
7,164 | 1 = \frac{2}{-\frac{1}{3 - \frac{2}{(-1) + 3}}\cdot 2 + 3} |
10,015 | r = \frac{r}{2} + \dfrac14 \cdot (1 - r) \Rightarrow \dfrac13 = r |
11,377 | 21/8 = 9/8 + 1/2 + 4/4 |
47,354 | D^c = D^c |
-20,353 | -\dfrac{27}{-36} = 3/4 (-\dfrac{9}{-9}) |
15,281 | \frac{y}{N} + z = (y + N*z)/N |
22,682 | d + \xi + f = f + d + \xi |
16,558 | x = \frac{d_1}{d_1 + d_2} \Rightarrow \frac{d_2}{d_1 + d_2} = -x + 1 |
22,387 | {6 \choose 1} {5 \choose 1} {9 \choose 3} = 2520 |
-10,190 | 25\% = \tfrac{25}{100} = 0.25 |
2,018 | 3 = \binom{\left(-1\right) + 2 + 2}{2 + \left(-1\right)} |
-1,129 | -\dfrac12\cdot 8/1 = (\frac{1}{2}\cdot (-1))/(1/8) |
37,809 | 2^n\cdot 2 = 2^{n + 1} |
20,314 | 44 \times x^5 = x \times 2 \times x^4 \times 9 + 6 \times x^5 + 5 \times x \times x \times x \times x^2 \times 4 |
-1,820 | \dfrac{1}{12} \cdot 23 \cdot π + π/4 = \frac16 \cdot 13 \cdot π |
8,351 | \left(f + h_2\right)/(h_1) = \frac{h_2}{h_1} + \frac{f}{h_1} |
3,019 | \frac{n^2 + 2\cdot n}{(n + 1)^2} = \frac{n}{(1 + n)^2}\cdot (n + 2) |
34,571 | A*x = x*A |
6,569 | -x^2 + y^2 = (-x + y) \cdot \left(y + x\right) |
21,111 | 2^{\left(-1\right) + n} \cdot 2^{\left(-1\right) + n}\cdot 2^{-n} = 2^{n + 2\left(-1\right)} |
4,990 | (n - k)*(n - k + (-1))! = \left(n - k\right)! |
14,405 | \frac13 \cdot (-y + x) = 1 \Rightarrow 3 = x - y |
16,263 | 1 = \frac1a + \frac{1}{a + b} + \dfrac{1}{a + b + c} \geq \frac{1}{a + b + c} \cdot 3 |
42 | -m * m*(6*20 + 30 \left(-1\right)) = -m^2*90 |
30,491 | 84\cdot 90 = \left(87 + 3\cdot (-1)\right)\cdot \left(87 + 3\right) = 87^2 - 3^2 = 7569 + 9\cdot \left(-1\right) = 7560 |
20,221 | \dfrac{1}{5^{\tfrac{1}{4}}} = \frac{5^{3/4}}{5} |
15,200 | 0 = 4 + 2 \times z, 0 = 4 \times (-1) + i \times 2 \implies [-2, 2] = \left[z,i\right] |
-20,848 | \frac{30 x + 25}{-x*36 + 30 (-1)} = -5/6 \frac{1}{-6 x + 5 (-1)} (-6 x + 5 (-1)) |
-6,102 | \frac{1}{\left(4(-1) + q\right)*2} = \frac{1}{8\left(-1\right) + 2q} |
6,698 | \left\lceil{\dfrac{1}{-\frac18 + \pi + 3*(-1)}}\right\rceil = 61 |
39,529 | (x\cdot H_1)^T\cdot H_2 = H_1^T\cdot x^T\cdot H_2 = H_1^T\cdot x\cdot H_2 |
8,032 | 3 \times 2^k - 2 \times 2^{k + (-1)} = 3 \times 2^k - 2^k = 2^k \times (3 + (-1)) = 2^k \times 2 = 2^{k + 1} |
-20,623 | -6/(-6)\cdot \left(-9/4\right) = 54/(-24) |
-20,887 | \frac{1}{25} \cdot (-p \cdot 20 + 30) = \left(6 - 4 \cdot p\right)/5 \cdot \frac55 |
1,311 | \frac{1}{{m \choose j + (-1)}}\cdot {m \choose j} = (m - j)/j = \frac{m}{j} + (-1) |
11,086 | -\dfrac{1}{8} + \dfrac{3}{16} = 1/16 |
30,063 | \frac{d}{dx} x^r = \lim_{w\to x} \frac{w^r-x^r}{w-x} = \lim_{w\to x}\frac{(w-x)(w^{r-1}+w^{r-2}x + w^{r-3}x^2 + \cdots + x^{r-1})}{w-x} |
20,784 | 1 = -b + h \Rightarrow 1 = h,0 = b |
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