physics and mathematics: Algebraic Formulas

Monday, 9 July 2012

Algebraic Formulas

$a^2-b^2 = (a+b)\times (a-b)$

$a^2+b^2 = (a+b)^2-2ab=(a-b)^2+2ab$

$(a+b)^2 = a^2+2ab+b^2 = (a-b)^2+4ab$

$(a-b)^2 = a^2-2ab+b^2 = (a+b)^2-4ab$

$a^3+b^3 = (a+b) \times (a^2-ab+b^2) = (a+b)^3-3ab \times (a+b)$

$a^3-b^3 = (a-b) \times (a^2+ab+b^2) = (a-b)^3+3ab \times (a-b)$

$(a+b)^3 = a^3+3a^2b+3ab^2+b^3 = a^3+b^3+3ab \times (a+b)$

$(a-b)^3 = a^3-3a^2b+3ab^2-b^3 = a^3-b^3-3ab \times (a+b)$

$(x+a) \times (x+b) = x^2+x \times (a+b)+ab$

$(x-a) \times (x+b) = x^2+x \times (b-a)-ab$

$(x-a) \times (x-b) = x^2-x \times (a+b)+ab$

$(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ac$

$(a+b+c)^3 = a^3+b^3+c^3+3(a+b)(b+c)(c+a)$

$a^4-b^4 = (a-b) \times (a+b) \times (a^2+b^2)$

$a^6-b^6 = (a+b) \times (a^2-ab+b^2) \times (a-b) \times (a^2+ab+b^2)$

$a^6+b^6 = (a^2+b^2) \times (a^4-a^2b^2+b^4)$

$a^4+a^2b^2+b^4 = (a^2-ab+b^2) \times (a^2+ab+b^2)$

$(a-b-c)^2 = a^2+b^2+c^2-2ab+2bc-2ac$

$a^3+b^3+c^3-3abc = (a+b+c) \times (a^2+b^2+c^2-ab-bc-ac)$

Indices Formulas:

The formulas involving relations between variables and their powers or powers and indices are:

$x^m \times x^n = x^{m+n}$

$x^m \times x^n \times \ldots \times x^p = x^{m+n+ \ldots +p}$

$x^m \div x^n = x^{m-n}$

$x^m \div x^n \div \ldots \div x^p = x^{m-n- \ldots -p}$

$(x^m)^n = x^{m \times n}$

$((x^m)^n)^o) = x^{m \times n \times o}$

$x^0 = 1$

$x^{-m} = \dfrac{1}{x^m}$

$x^{m} = \dfrac{1}{x^{-m}}$

$x^{\frac{m}{n}} = \sqrt[n]{x^m}$

$\left( \dfrac{x^a}{y^b} \right)^c = \dfrac{x^{ac}}{y^{bc}}$

$\dfrac{x^m}{y^m} = \left( \dfrac{x}{y} \right)^m$

$\sqrt[m]{\dfrac{x^a}{y^b}} = \dfrac{x^{\frac{a}{m}}}{y^{\frac{b}{m}}}$

$x^{\frac{p}{q}} = \sqrt[q]{x^p} = \left(\sqrt[q]{x}\right)^p$

$\sqrt[m]{\dfrac{x}{y}} = \dfrac{\sqrt[m]{x}}{\sqrt[m]{y}}$

$\sqrt{a} \times \sqrt {b} = \sqrt{a \times b}$

If , $a^x = a^y$ then , x=y. ( Provided That : $0 < a\text{ and }a \ne 1$ )

If, $a^x = b^x$ then , a=b. ( Provided That : $0 < a , b \text{ and }a , b \ne 1$ )

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