id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
10,996 |
-\cos^2(x^2) + 1 = \sin^2(x^2)
|
34,195 |
(\frac{1}{(-1) + u} - \dfrac{1}{1 + u})/2 = \tfrac{1}{u \cdot u + (-1)}
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39,265 |
\cos(i\varepsilon) = \frac{1}{2}(e^{-\varepsilon} + e^\varepsilon) = \cosh\left(\varepsilon\right)
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24,345 |
\frac{1}{\binom{n}{p + (-1)}}*\binom{n}{p} = \frac{n!*\frac{1}{p!*(n - p)!}}{n!*\dfrac{1}{(p + (-1))!*(n - p + (-1))!}} = \frac{(p + \left(-1\right))!}{p!*(n - p)!}*(n - p + (-1))! = \dfrac{(p + (-1))!}{p!*(n - p)!}*\left(n - p + 1\right)! = (n - p + 1)/p
|
27,617 |
-\binom{9}{3} + \binom{15}{3} = \binom{6 + 5 + 4}{3} - \binom{4 + 5}{3}
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-2,595 |
24^{1 / 2} + 96^{\frac{1}{2}} - 6^{1 / 2} = \left(16*6\right)^{1 / 2} - 6^{1 / 2} + \left(4*6\right)^{1 / 2}
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10,132 |
{7 \choose 1} \cdot {6 \choose 2} \cdot {4 \choose 4} = \frac{7!}{1! \cdot 4! \cdot 2!}
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-20,914 |
\frac{1}{(-25) y}\left(45 - 30 y\right) = (-6y + 9)/((-5) y) \frac{1}{5}5
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30,446 |
e^{6 z} z = 1 \implies 0.2387 \approx z
|
-5,207 |
1.56\times 10 = \dfrac{1.56}{1000}\times 10 = 1.56/100
|
21,165 |
(\frac1x)^{1/5} = (\frac{1}{x})^{1/5} = x^{-1/5}
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-19,473 |
\frac{1}{3 \cdot \tfrac18 \cdot 7} = \frac{8 / 7}{3} \cdot 1
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14,592 |
1/2 \cdot 1/2 \cdot 1/2 = 1/8
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12,002 |
k \cdot 4 + 1 = 1 + k \cdot 4 + \left(-1\right) + 1
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-7,912 |
(14 + 34 \times i - 21 \times i + 51)/13 = (65 + 13 \times i)/13 = 5 + i
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-25,784 |
\frac{11}{48} = 11\cdot \frac{1}{12}/4
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-5,440 |
1.8 \times 10^{-3\,+\,-5} = 1.8 \times 10^{-8}
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9,328 |
\mathbb{E}(A^2\cdot Y^2) = \mathbb{E}(A^2)\cdot \mathbb{E}(Y \cdot Y)
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606 |
\frac{1}{1 + 2 \cdot a^2 \cdot c} = 1 - \dfrac{2 \cdot a^2 \cdot c}{1 + 2 \cdot a \cdot a \cdot c} \geq 1 - \frac{a \cdot c}{\sqrt{2 \cdot c}} \cdot 1
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31,352 |
g\cdot \cos^2(d) = -\cos(d) + b \Rightarrow g\cdot \cos^2(d) + \cos\left(d\right) - b = 0
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14,320 |
1 = x^2 rightarrow x = 1
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16,259 |
(4/6)^5 = (\tfrac23)^5
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10,351 |
\tfrac{1}{x^{\frac13}}x^{\frac{4}{3}} = x^{\tfrac43 - \dfrac13} = x^{\frac133} = x
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-15,774 |
\frac{1}{10}\cdot 5 - 6\cdot \frac{9}{10} = -49/10
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10,325 |
9 - 6\cdot 3 = -9
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-16,863 |
4 = 4(-4j) + 4 \cdot 6 = -16 j + 24 = -16 j + 24
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-14,939 |
336 = 78 + 83 + 88 + 87
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-1,096 |
\frac{5}{8}*\frac59 = \dfrac{5*1/9}{8*1/5}
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6,717 |
1 - \cos{4\cdot y} = 2\cdot \sin^{22}{y} = 8\cdot \sin^2{y}\cdot \cos^2{y} = 8\cdot (1 - \cos{y})\cdot (1 + \cos{y})\cdot \cos^2{y}
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2,552 |
T \times E \times B = B \times T \times E
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-26,390 |
\epsilon^{k + l} = \epsilon^l*\epsilon^k
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73 |
{n + \left(-1\right) \choose 2*i + (-1)} = {n + (-1) \choose n + (-1) - 2*i + (-1)} = {n + (-1) \choose n - 2*i}
|
10,427 |
(-1) + Q^x = ((-1) + Q)\cdot (1 + Q + \dotsm + Q^{(-1) + x})
|
-4,698 |
x^2 + x*5 + 6 = (2 + x)*\left(x + 3\right)
|
16,543 |
(3^{y - z} + 1)*3^z = 3^z + 3^y
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21,780 |
1 = 20*z + 3*y rightarrow z = -1, y = 7
|
19,147 |
2 \cdot \sin(X) \cdot \cos(X) = \sin(2 \cdot X)
|
-21,035 |
\frac{1}{28 \cdot (-1) - 35 \cdot f} \cdot (42 \cdot \left(-1\right) + f \cdot 7) = \dfrac{f + 6 \cdot (-1)}{4 \cdot (-1) - 5 \cdot f} \cdot 7/7
|
3,846 |
\left(\sqrt{f} \cdot \sqrt{g}\right)^2 = f \cdot g
|
11,056 |
JK + IK = K\cdot (J + I)
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15,118 |
\dfrac{1 + 2\cdot n}{(-1) + n\cdot 2}\cdot 2 = \frac{1}{1 + n}\cdot \left(2\cdot n + 1\right)\cdot \frac{n\cdot 2 + 2}{n\cdot 2 + (-1)}
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12,127 |
183 - \left(-1\right) + k = 184 - k
|
-184 |
\frac{8!}{(8 + 4\cdot (-1))!\cdot 4!} = {8 \choose 4}
|
2,642 |
128 = \tfrac{256}{2}
|
-11,491 |
-15*i - 23 = -25 + 2 - i*15
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10,564 |
|7 + 2\times (-1)| = 5 > 2
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-1,404 |
(\left(-5\right)*1/3)/(1/4*5) = -\frac53*4/5
|
22,299 |
\cos(2\cdot s) = 1 - \sin^2(s)\cdot 2
|
13,434 |
-f\cdot \left(-b\right) = fb
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-9,588 |
0.01\cdot \left(-87\right) = -87.5/100 = -\frac{1}{8}\cdot 7
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8,818 |
\tfrac{1}{30^4} \cdot 657720 = 203/250
|
2,552 |
C*Z*B = C*Z*B
|
-9,354 |
y \cdot 11 + 66 = y \cdot 11 + 2 \cdot 3 \cdot 11
|
-1,910 |
\pi/3 - \dfrac{11}{12} \pi = -\frac{7}{12} \pi
|
-3,220 |
\sqrt{6} \cdot 4 - 2 \cdot \sqrt{6} = -\sqrt{4} \cdot \sqrt{6} + \sqrt{16} \cdot \sqrt{6}
|
15,184 |
|y^2 - y| = |y|*|(-1) + y|
|
10,340 |
d^3 = d^2 d
|
31,620 |
\sqrt{\tfrac{x + 1}{(-1) + x}} = u \implies x = \dfrac{u^2 + 1}{u^2 + (-1)} = 1 + \frac{1}{u^2 + (-1)}2
|
-15,781 |
-\dfrac{5}{10} + 1 = 5/10
|
-64 |
(-1) + 1 = 0
|
14,448 |
\pi e^x = d/dx (\pi e^x)
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19,132 |
3 (-1) + y \geq 0 rightarrow 3 \leq y
|
23,814 |
0 = -\frac{4}{u^3} - 4/u - u + \dfrac{3*b}{u^2}*1 = \frac{1}{u^3}*(-4 - 4*u^2 - u^3 + 3*u*b)
|
35,412 |
12 = 2! \cdot 3!
|
14,000 |
\frac{1}{2! \cdot 3!} 7! = 420
|
29,831 |
e^{-z^2} y = a + z \Rightarrow y = e^{-z^2} a + ze^{-z^2}
|
4,798 |
b \cdot a \cdot b = b \cdot a \cdot b
|
-26,419 |
1/\left(390625\times 9765625\right) = 5^{-8 - 10} = 5^{-8 + 10\times (-1)} = 1/3814697265625
|
-20,780 |
\tfrac{1}{-63\cdot z + 42\cdot (-1)}\cdot (45\cdot z + 30) = -5/7\cdot \tfrac{-9\cdot z + 6\cdot (-1)}{-9\cdot z + 6\cdot (-1)}
|
3,889 |
\frac{4^t + (-1)}{1 + 2*t} = \frac{1}{2*t + 1}*((-1) + 2^t)*\left(1 + 2^t\right)
|
8,292 |
\cos{z} = \frac12(e^{iz} + e^{-iz}) \sin{z} = \frac{1}{2i}(e^{iz} - e^{-iz})
|
-18,070 |
84 + 18*(-1) = 66
|
23,040 |
50 \cdot z^2 - 50 \cdot z + 25 = 25 + 50 \cdot z^2 - z \cdot 50
|
45,695 |
y = |y| = (y^2)^{1/2}
|
19,744 |
d - b = \frac{-b^2 + d^2}{b + d}
|
20,419 |
z y = 1/(z y) = \dfrac{1}{y z} = y z z
|
29,709 |
27000 = 3^3\times 2 \times 2^2\times 5 \times 5 \times 5
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-8,537 |
8/6 - 10^{-1} = \frac{5}{6\cdot 5}\cdot 8 - 3/(10\cdot 3) = \frac{1}{30}\cdot 40 - \frac{3}{30} = \dfrac{1}{30}\cdot (40 + 3\cdot (-1)) = \frac{37}{30}
|
3,937 |
(-\sqrt{3} + 2) \times 5 = 10 - \sqrt{3} \times 5
|
12,538 |
3 \cdot (30 + 20) = 150
|
7,809 |
3 = 128 + 125*\left(-1\right) = 2^7 - 5^3
|
-10,611 |
\frac{5}{5}\cdot (-\frac{1}{t + 3\cdot \left(-1\right)}\cdot 6) = -\frac{30}{5\cdot t + 15\cdot (-1)}
|
29,246 |
(-d + x) \cdot (d + x) = x \cdot x - d^2
|
3,675 |
\left(SA\right)^T B = A^T S^T B = A^T SB
|
-595 |
(e^{\frac{\pi}{12}\cdot i})^{13} = e^{13\cdot \frac{\pi\cdot i}{12}}
|
25,024 |
c^n\cdot c^l = c^{n + l}
|
8,524 |
( h, c_b\cdot f') = c_b\cdot f'\cdot h
|
-2,569 |
\sqrt{9} \cdot \sqrt{11} - \sqrt{4} \cdot \sqrt{11} = 3\sqrt{11} - 2\sqrt{11}
|
10,957 |
(\frac{1}{2})^{1/8} = 2^{7/8}/2
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-15,431 |
\frac{1}{z^4\cdot \dfrac{1}{x^2}}\cdot z^3 = \frac{z^3}{\frac{1}{x^2}}\cdot \frac{1}{z^4} = x \cdot x/z = \frac{x^2}{z}
|
20,158 |
m n = m n
|
34,584 |
6058655748 = 61 \cdot 61^2 + 1823^3 = 1049^3 + 1699^3
|
-10,001 |
\frac{5}{10} = 1/2
|
27,625 |
x = \dfrac{\pi}{4} \Rightarrow \cos^5\left(x\right) - \sin^5(x) = 0 \neq \cos\left(5\cdot \pi/4\right)
|
21,007 |
32 = (1 + 3) \left(3 + 1\right) (1 + 1)
|
-20,669 |
\frac{4*(-1) - r*3}{30*r + 40} = -1/10*\frac{1}{r*3 + 4}*(3*r + 4)
|
1,916 |
(1 + n \cdot n \cdot n) (n^3 + \left(-1\right)) = n^6 + (-1)
|
21,171 |
v*(h + d) = dv + vh
|
4,074 |
1/(d\cdot c) = 1/(d\cdot c)
|
20,827 |
X_m\cdot X_g\cdot f = m\cdot g\cdot f = g\cdot m\cdot f = X_g\cdot X_m\cdot f
|
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