Documents/Small/mat1/mat1.tex

\documentclass[11pt,a4paper,leqno]{article}

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\usepackage{amsmath}
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\begin{document}
\noindent
\setlength\parindent{0pt}
\title{asd}

\section{Radicals}

$\sqrt{8} = \sqrt{4}\sqrt{2} = 2\sqrt{2}$

$\sqrt[3]{8} = \sqrt[3]{8}\sqrt[3]{2} = 2\sqrt[3]2$

$\sqrt[4]{32} = \sqrt[4]{16}\sqrt[4]{2} = 2\sqrt[4]2$

$\sqrt{x^3} = \sqrt{x^2}\sqrt{x} = |x|\sqrt{2}$

$\sqrt{x^7} = \sqrt{x^6}\sqrt{x} = x^{\frac{6}{2}} = x^3\sqrt{2}$

$\sqrt[3]{x^13} = \sqrt[3]{x^{12}}\sqrt{x} = x^{\frac{12}{3}} = x^4\sqrt[3]{x}$

$\sqrt{50x^3y^5} = \sqrt{25}\sqrt{2}\sqrt{x^2}\sqrt{x}\sqrt{y^4}\sqrt{y} = 5xy^2\sqrt{2xy}$

$\frac{8}{\sqrt{3}} = \frac{8}{\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}} = \frac{8\sqrt{3}}{3}$

$\frac{7}{\sqrt[3]{4}} = \frac{7}{\sqrt[3]{4}} * \frac{\sqrt[3]{4^2}}{\sqrt[3]{4^2}} = \frac{7\sqrt[3]{16}}{4} = \frac{7 * 2\sqrt[3]{2}}{4} = \frac{7\sqrt[3]{2}}{2}$

Here should be more Radicals + Exponents

$\sqrt[3]{x^7} * \sqrt[5]{x^3} = x^{\frac{7}{3}} * x^{\frac{3}{5}} = x^{\frac{7}{3} + \frac{3}{5}} = x^{\frac{35}{15} + \frac{9}{15}} = x^{\frac{44}{15}} = x^{\frac{30}{15}} * x^{\frac{14}{15}} = x^2 * x^{\frac{14}{15}} = x^2\sqrt[15]{x^{14}}$

$\frac{\sqrt[4]{x^9}}{\sqrt[3]{x^2}} = \frac{x^{\frac{9}{4}}}{x^{\frac{2}{3}}} = \frac{x^{\frac{27}{12}}}{x^{\frac{8}{12}}} = x^{\frac{19}{12}} = \sqrt[12]{x^{19}} = x\sqrt[12]{x^7}
 $

\section{test}

$5x + 4x = 9x$

$3x + 4y + 5x + 8y = 8x + 12y$

$3\sqrt{2} + 5\sqrt{7} + 8\sqrt{2} + 3\sqrt{7} = 11\sqrt{2} + 8\sqrt{7}$

$7x + 4x^2 + 5x + 9x^2 = 13x^2 + 12x$

$(9x^2 + 6x + 5) + (3x^2 - 5x - 9) = 12x^2 + x - 4$

$(3x^2 + 7x - 4) - (5xˇ2 - 5x + 7) = 3x^2 + 7x - 4 -8x^2 + 5x - 7 = -5xˇ2 + 12x -11$ 


$7x(x^2 + 2x -3) = 7x^3 + 14x^2 -21x$

$x^1 * x^2 = x^3$

$(5x^2)(3x^4 - 6x^3 + 5x - 8) = (5 * 3)x^{2 + 4} - (5 * 6)x^{2 + 3} + (5 * 5)x^{1 + 2} - (5 * 8)x^2 = 15x^6 - 30x^5 + 25x^3 - 40x^2$

$(3x - 4)(2x + 7) = (3  * 2)(x * x) + (3 * 7)x -(4 * 2)x - (4 * 7) = 6x^2 + 21x - 8x - 28 = 6x^2 + 13x - 28$

$(2x - 5)(4x + 7) = 8x^2 + 14x - 20x - 35 = 8x^2 - 6x - 35$

$(2x - 3)^2 = (2x-3)(2x-3) = 4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9$

$(5x - 9)(2x^2 - 3x + 4) = 10x^3 -15x^2 + 20x -18x^2 +27x -36 = 10x^3 -33x^2 + 47x -36$

\subsection{Exponents}

$x^3 * x^4 = x^{3 + 4} = x^7$

$x^2 * x^3 = (x * x) * (x * x * x) = x^5$

$\frac{x^9}{x^4} = x^{9 - 4}$

$\frac{x^5}{x^2} = \frac{x * x * x * x * x}{x * x} = \frac{\cancel{x * x} * x * x * x}{\cancel{x * x}} = \frac{x * x * x}{1} = x^3$

$\frac{x^4}{x^7} = \frac{x * x * x * x}{x * x * x * x * x * x * x} = \frac{\cancel{x * x * x * x} * x * x * x}{\cancel{x * x * x * x} * x * x * x} = \frac{1}{x * x * x} = \frac{1}{x^3} = x^{-3}$

$(x^7)^6 = x^{7 * 6} = x^{42}$

$(x^2)^3 = (x^2) * (x^2) * (x^2) = (x * x) * (x * x) * (x * x) = x^6$

\subsection{Moar}

$(3x^4y^5)(5x^6y^7) = (3 * 5)(x^4 * x^6)(y^5 * y^7) = 15x^{(4 + 6)}y^{(5 + 7)} = 15x^{10}y^{12}$

$(8x^3y^{-2})(7x^{-8}y^5) = 56x^{-5}y^3 = \frac{56y^3}{x^5}$

$\frac{24x^7y^{-2}}{6x^4y^5} = \frac{4x^{7 - 4}y^{-2 - 5}}{1} = \frac{4x^{3}y^{-7}}{1} = \frac{4x^{3}}{x^7}$

$\frac{32x^5y^{-3z^4}}{40x^{-8}y^{-7}z^{-8}} = \frac{4x^{5 - (-8)}y^{-3 - (-7)}}{5} = \frac{4x^{13}y^4z^{12}}{5}$


$(3x^3)^2 = (3^1x^3)^2 = 3^{1 * 2}x^{3 * 2} = 3^2x^6 = 9x^6$

$(4x^2y^3)^3 = 4^{1 * 3}x^{2 * 3}y^{3 * 3} = 4^3x^6y^9 = 64x^6y^9$

$-2(5xy^3)^0 = -2 * (5^{1 * 0}x^{1 * 0}y^{3 * 0}) = -2 * (5^0x^0y^0) = -2 * (1 * 1 * 1) = -2$

$\frac{5x^{-2}}{y^{-3}} * \frac{8x^4}{y^{-5}} = \frac{5y^3}{x^2} * \frac{8x^4y^5}{1} = \frac{40y^8x^2}{x^2} = 40y^8$

$\frac{35x^{-3}}{40xy^5} * \frac{24x^2y^2}{42y^{-4}} = \frac{\cancel{7} * \cancel{5}}{\cancel{8} * \cancel{5} * x * x^3 * y^5} * \frac{\cancel{8} * \cancel{3} * x^2 * y^2 * y^4}{\cancel{7} * \cancel{3} * 2} = \frac{x^2y^6}{2x^4y^5} = \frac{y}{2x^2}$

$\frac{24xy}{27x^{-2}} / \frac{36x^2y^{-3}}{45xy^4} = \frac{24xy}{27x^{-2}} * \frac{45xy^4}{36x^2y^{-3}} = ...$


$\frac{\frac{3x}{5}}{\frac{7xy}{9}} = \frac{3x}{5} / \frac{7xy}{9} = \frac{3x}{5} * \frac{9}{7xy} = \frac{27}{35y}$

\subsection{Equations}

Only a few interesting cases

$(0.04x + 0.15 = 0.09x - 0.025) / * 100 $

$4x + 15 = 9x - 25$

$x = 8$

----

$x^2 - 25 = 0$

$x^2 = 25$

$\sqrt{x^2} = \sqrt{5}$

$x = \pm 5$


\subsection{Factorization}


$x^2 - 25 = (x + 5)(x - 5)$

$2x^2 - 18 = 2(x^2 - 9) = 2(x - 3)(x + 3)$

$x^4 - 81 = (x^2 - 9)(x^2 + 9)$

$x^2 - 5x + 6 = (x - 2)(x - 3)$

$x^2 - 2x - 15 = (x + 3)(x - 5)$

$2x^2 + 3x - 2 = 2x^2 + 4x - 1x - 2 = 2x(x + 2) - 1(x + 2) = (x + 2)(2x - 1)$


$x^3 - 4xˇ2 - x + 4 = x^2(x - 4) -1(x - 4) = (x - 4)(x^2 - 1)$

\subsection{Other}

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

\end{document}