Algebric Identities
$$(a + b + c) x = ax + bx + cx $$
$$\frac { a + b + c }{x} = \frac{a}{x} + \frac{b}{x} + \frac{c}{x}$$
On the Basis of number of terms:
On the basis of degree:
$$\frac {a}{b} + \frac {c}{d} = \frac{ad + bc}{bd}$$
$$\frac {a}{b} - \frac {c}{d} = \frac{ad - bc}{bd}$$
$$\frac {a}{b} \times \frac {c}{d} = \frac{ac}{bd}$$
$$(a + b)^2 = a^2 + b^2 + 2ab $$
$$(a - b)^2 = a^2 + b^2 - 2ab $$
$$(a^2 - b^2) = (a + b)(a - b)$$
$$ (x + a)(x +b) = x^2 + (a + b)x + ab $$
$$(a + b + c)^2 = a^2 + b^2 + c ^2 + 2ab + 2bc + 2ca $$
$$(a + b - c)^2 = a^2 + b^2 + c ^2 + 2ab - 2bc - 2ca $$
$$(a - b - c)^2 = a^2 + b^2 + c ^2 - 2ab + 2bc - 2ca $$
$$(a + b)^3 = a^3 + b^3 + 3a^b + 3ab^2 $$
$$(a - b)^3 = a^3 - b^3 - 3a^b + 3ab^2 $$
$$ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 -ab -bc -ca) $$
$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$
$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$
$$ a^m \times a^n = a^{m+n}$$
$$ (a.b)^m = a^m . a^n$$
$$ a^0 = 1, a \neq 0$$
$$ (a^m)^n = a^{m.n}$$
$$ \frac{a^m}{a^n} = a^{m - n}$$
$$ (\frac{a}{b})^m = \frac{a^m}{b^m}$$
$$ a^{-m} = \frac{1}{a^m}a \neq 0$$
$$a^{\frac{1}{n}} = \sqrt [ n ]{ a } $$
$$a^{\frac{m}{n}} = \sqrt [ n ]{ a^m } $$
$$\sqrt [ n ]{ a } = a^{\frac{1}{n}}$$
$$\sqrt [ n ]{ a^m } = a^{\frac{m}{n}}$$
$$\sqrt[n]{a.b} = \sqrt[n]{a}.\sqrt[n]{a}$$
$$\frac{\sqrt[n]{a}}{\sqrt[m]{b}} = \sqrt[nm]{\frac{a^m}{b^n}}$$