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math_formulas

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1 + x^4 = 1 + 2 \cdot x^2 + x^4 - 2 \cdot x^2 = (1 + x^2) \cdot (1 + x^2) - (2^{\frac{1}{2}} \cdot x)^2

2^{a + g} = 2^g \cdot 2^a

a*2 + b*7 + 3 c = 2 (2 b + a) + (c + b)*3

-21/24 = \frac{(-21) \cdot 1/3}{24 \cdot \frac{1}{3}} = -\tfrac18 \cdot 7

7/100 + 171/200 = \frac{1}{200}\cdot 185 = 0.925

{k + (-1) + 4 \choose k + (-1)} = {k + 3 \choose k + \left(-1\right)} = {k + 3 \choose 4}

-23/12*\pi + \pi*5/6 = -13/12*\pi

((-1) + l) \cdot 2 = 2 \cdot (-1) + l \cdot 2

\left(-1\right) + (1 + a) \cdot (1 + b) = a \cdot b + a + b

e^r = x,b = e^k rightarrow e^{r + k} = b\cdot x

\frac{36}{72} = \frac{36\cdot \dfrac{1}{36}}{72\cdot \frac{1}{36}} = 1/2

\tfrac{2}{5 m + 25 (-1)} = \frac{2}{(5 \left(-1\right) + m)*5}

\frac{1}{29}(100 - 105 i - 40 i + 42 (-1)) = \frac{1}{29}(58 - 145 i) = 2 - 5i

(S + (-1))\cdot 2 + 1 = 2S + (-1)

\frac{8}{10} + \frac{6}{100} = 80/100 + \frac{6}{100}

\nu_T = \nu_T

3/5*\frac{1}{5}*4 = \frac{12}{25}

88 = 2^3 \cdot 11

28 = 64 + 48*f + 12*f^2 + f^3 - f * f * f = 64 + 48*f + 12*f * f

\dfrac19\cdot 2 = s/(12\cdot \pi)\cdot 12\cdot \pi = s

-2\cdot 3 + \dfrac{1}{3}\cdot 3^3 = 3

\left(k + 2\right)*(1 + k)! = (k + 2)!

-\frac{17}{6} - \frac{1}{3}\cdot 14 = -7.5

50 = 5^2*2 + 0^2

(1 + 2x) (x \cdot 2 + (-1)) + k \cdot 4 + 1 = 4(k + x^2)

x \cdot x - 14\cdot x + 58 = x^2 - 14\cdot x + 49 + 9 = (x + 7\cdot (-1)) \cdot (x + 7\cdot (-1)) + 9 = (x\cdot (-7)) \cdot (x\cdot (-7)) + 3^2

-k - l = -(k + l)

\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5}

A = A^{1/2}*A^{1/2}

\frac{\frac1h}{x}\cdot g = 1/h\cdot g/x

\cos(\dfrac{\pi}{2} - v) = \sin{v}

\tfrac{1}{z} \cdot z^2 = z

(a + d_2 + d_1)\times \left(a^2 + d_2 \times d_2 + d_1^2 - a\times d_2 - d_2\times d_1 - d_1\times a\right) = -3\times a\times d_1\times d_2 + a^3 + d_2^3 + d_1^3

\dfrac{15}{7 + 8} = \frac{1}{15}*15 = \frac{15}{15} = 1

\ln(5) + \ln(400 - t) = \ln(-t \times 5 + 2000)

-55/10 = -10 \cdot \frac{7}{10} + 3/10 \cdot 5

\tfrac{x^2 + 1}{x + 1} - x = \frac{1}{x + 1}(x^2 + 1 - x^2 - x) = \dfrac{(-1) x}{x + 1}

\frac{a^2 + a}{a^2 - 3\cdot a + 4\cdot (-1)} = \frac{(a + 1)\cdot a}{(a + 1)\cdot (4\cdot (-1) + a)}

\mathbb{E}[W^3] = \mathbb{E}[-W]^3 = -\mathbb{E}[W^3]

\frac{1}{2\cdot \sqrt{3 + 5\cdot z}}\cdot (75\cdot z^2 + 36\cdot z) = \frac{d}{dz} (3\cdot \sqrt{5\cdot z + 3}\cdot z^2)

2*3^2*5*11 = 990

\dfrac{n^{20}\cdot q^{15}}{q^{15}\cdot \frac{1}{n^9}} = \frac{1}{\left(\frac{q^5}{n^3}\right) \cdot (\frac{q^5}{n^3})^2\cdot \frac{1}{n^{20}\cdot q^{15}}}

-4^3 = \left(-4\right) \left(-4\right) (-4)

\dfrac78 = 1/8 + \frac78 - \dfrac18

-2^4 + 3^3 = 11

\sqrt{3} - \sqrt{3}*4 + 5\sqrt{3} = \sqrt{3} - \sqrt{3} \sqrt{16} + \sqrt{25} \sqrt{3}

\frac{-z\times 7 + 2}{-z\times 7 + 2}\times (-6/5) = \dfrac{1}{10 - 35\times z}\times (12\times (-1) + 42\times z)

2\cdot \sqrt{5} + \sqrt{5}\cdot 3 = \sqrt{5}\cdot \sqrt{4} + \sqrt{9}\cdot \sqrt{5}

\frac{5}{5}*\dfrac{5}{4y + 20} = \frac{25}{20 y + 100}

f = \tfrac{1}{1/f}

6^{1 / 2}\times (1 + 3 + 2) = 6\times 6^{1 / 2}

\frac{1}{10} \times 4 \times 10 - 6 \times 6/10 = 4/10

\dfrac{48 x + 72 \left(-1\right)}{45 - 30 x} = -\frac{1}{5}8 \tfrac{9 - 6x}{-x*6 + 9}

492 = (\binom{14}{7} + 2 \cdot (-1))/7 + 2

10/10 = 10 \cdot 5/10 - 8 \cdot 5/10

\operatorname{E}(x^2) - \operatorname{E}(x)^2 = \operatorname{E}((-\operatorname{E}(x) + x)^2)

(-1) + x^2 = \left(x + 1\right) (x + (-1))

(\left(33 + 29 \cdot (-1)\right)^2 + (43 \cdot (-1) + 33)^2 + (7 + 0 \cdot \left(-1\right))^2 + (21 \cdot (-1) + 33)^2)/4 = 77.25

\frac{1}{2}\times \infty = \infty

y^2 - 2\cdot i = y^2 - (1 + i)^2 = (y + (-1) - i)\cdot (y + 1 + i)

1 - \frac{332}{998} = \frac{1}{998}\cdot 666

\cos{z} = \sin(\tfrac{\pi}{2} - z)

4/36 = \tfrac19

\sin(\sin^{-1}{x}) + \sin(\sin^{-1}{x} + 2\pi) = x + x = 2x

\frac{1}{45} \cdot 3240 = 72

\cos(-F + \gamma) = \sin(F)\cdot \sin\left(\gamma\right) + \cos(\gamma)\cdot \cos(F)

(1 + i) \cdot (1 + i) = 1 + i^2 + i \cdot 2

0 = \tfrac00

|a + fi| = \sqrt{\left(a + fi\right) (a - fi)} = \sqrt{a \cdot a + f^2}

11/60 = s/(20*\pi)*20*\pi = s

|\dfrac{1}{(x + 2)*4} (x + 2 (-1))| = \frac{1}{4 |x + 2|} |x + 2 (-1)|

\left(y + (-1)\right) \cdot \left(3 + y^2 \cdot 2 + 3 \cdot y\right) = y^3 \cdot 2 + y \cdot y + 3 \cdot (-1)

\left(G\cdot G^U\right)^U = (G^U)^U\cdot G^U = G\cdot G^U

(u + v)\cdot k = k\cdot v + k\cdot u

\|FQ\|_2^2 = (FQy)^T FQy = y^T Q^T F^T FQy

n\frac{1}{n^2}e^{\frac{1}{n}n} = \dfrac1ne

\frac{1}{2!\cdot 3!}\cdot 5! = {5 \choose 3}

3 \cdot 10^{1/2} = 90^{1/2}

100 - y^2*9 = (10 + 3y) \left(10 - 3y\right)

-9*3^{1/2} = -3*27^{1/2}

\frac1x \cdot x = x \cdot \frac{x}{x}/x = (x/x)^2

b\cdot H = H\cdot a \Rightarrow \frac{H}{a} = \frac{H}{b}

\sinh{2} = \frac{e^4 + (-1)}{2 \cdot e^2} \approx 243/67

{2 \choose 2} {8 \choose 1} = 8

\dfrac{1}{c_1} \cdot c_1 \cdot c_2 \cdot l = l \cdot c_1/(c_1) \cdot c_2 \cdot c_1/(c_1)

21 = \frac12\left(39 + 3\right)

k_1 \cdot h_2 \cdot k_2 \cdot h_1 = h_2 \cdot k_1 \cdot k_2 \cdot h_1

0 = A^{n\cdot 2} rightarrow 0 = A^n

\frac{5}{4\cdot q + 40} = \frac{5}{\left(10 + q\right)\cdot 4}

\sin(-x + a) = -\cos{a}\cdot \sin{x} + \sin{a}\cdot \cos{x}

\frac{1}{12}\cdot 19\cdot \pi = 0 + \pi\cdot 19/12

e^{\frac{π}{6}\cdot i\cdot 15} = (e^{π\cdot i/6})^{15}

-i*2 + 8 + 1 = 9 - 2i

286 = 2*(10 + 1)*(10 + 3) = 2*11*13

(-1) + x^4 = (x^2 + 1)\cdot (x + 1)\cdot (\left(-1\right) + x)

1 = 1 \cdot 1

-\frac{64}{81} = -\frac{2}{3} \cdot \frac{1}{27} \cdot 32

w_1\cdot (w_1 + w) = w_1^2 + w_1\cdot w \gt w_1 + w

Q = -Y + B \Rightarrow Q + Y = B

\left(2\cdot i + 2\cdot \left(-1\right)\right)\cdot 2^{i + (-1)} = 2\cdot (i + (-1))\cdot 2^i/2 = (i + (-1))\cdot 2^i