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%opening
\newcommand{\snovdn}{Koreni poljubnih stopenj }
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\title{%
\snovdn --- \stevilkadn. domača naloga
\\
\large Matematika, Gimnazija Bežigrad}
\author{\begin{tabular}{rl}
\textbf{Profesor:} & prof. Vilko Domajnko \\
\textbf{Avtor:} & Anton Luka Šijanec, 2. a
% \textbf{Avtor:} & Anton Luka Šijanec \\ & Member 2 \\ & Member 3
\end{tabular}}
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\begin{document}
\maketitle
\begin{abstract}
Ta dokument vsebuje navodila in rešitve domače naloge snovi \textit{\snovdn}pri matematiki, ki sem jih spisal sam.
Kjer je bilo potrebno izbrati neke poljubne naloge, sem jih vedno izbiral naključno.
\end{abstract}
\paragraph{Navodilo naloge} \textbf{listi}: \textit{Koreni poljubnih stopenj} / 1, ... 10 (izberi tri sode in tri lihe, povsod po en primer)
%\tableofcontents
\begin{enumerate}[label=\textbf{\arabic*.}]
\item Izračunaj brez pomoči kalkulatorja:
\begin{enumerate}[label=\textbf{\xslalph*})]
\setcounter{enumii}{1}
\item $$\sqrt[3]{12,5\cdot10^{-5}}=\sqrt[3]{\frac{25}{2}\cdot\frac{1}{10^5}}=\sqrt[3]{\frac{1}{8000}}=\frac{1}{20}$$
\end{enumerate}
\setcounter{enumi}{2}
\item Izračunaj in poenostavi:
\begin{enumerate}[label=\textbf{\xslalph*})]
\setcounter{enumii}{4}
\item $$a\cdot\frac{\left(\sqrt[4]{a}+\sqrt[4]{b}\right)^2+\left(\sqrt[4]{a}-\sqrt[4]{b}\right)^2}{a+\sqrt{ab}}
=a\cdot\frac{\sqrt{a}+2\sqrt[4]{ab}+\sqrt{b}+\sqrt{a}-2\sqrt[4]{ab}+\sqrt{b}}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}
=a\cdot\frac{2\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}=$$$$=\frac{a2}{\sqrt{2}}=2\sqrt{a}$$
\end{enumerate}
\item Poenostavi:
\begin{enumerate}[label=\textbf{\xslalph*})]
\setcounter{enumii}{7}
\item $$\frac{\sqrt[3]{x^2\cdot\sqrt[4]{xy^3}}}{\sqrt{x\sqrt{xy}}}
=\frac{\sqrt[3]{\sqrt[4]{x^2^4\cdot xy^3}}}{\sqrt{\sqrt{x^2\cdot xy}}}
=\frac{\sqrt[3]{\sqrt[4]{x^9y^3}}}{\sqrt{\sqrt{x^3y}}}
=\frac{\sqrt[12]{\left(x^3y\right)^3}}{\sqrt{\sqrt{x^3y}}}
=\frac{\sqrt[4]{x^3y}}{\sqrt[4]{x^3y}}
=1$$
\end{enumerate}
\setcounter{enumi}{7}
\item Zaprta kvadratna škatla z merami $\SI{6}{\deci\meter}\times\SI{9}{\deci\meter}\times\SI{30}{\deci\meter}$ ima enako površino kot kocka. Izračunaj dolžino kockinega roba (do milimetra natančno).
$$
\sqrt{
\frac{
2\left(\SI{30}{\deci\meter}\cdot\SI{6}{\deci\meter}\right)+
2\left(\SI{6}{\deci\meter}\cdot\SI{9}{\deci\meter}\right)+
2\left(\SI{9}{\deci\meter}\cdot\SI{30}{\deci\meter}\right)
}{6}
}
=\sqrt{\frac{\SI{1008}{\deci\meter\squared}}{6}}=\sqrt{\SI{168}{\deci\meter\squared}}=\SI{12,961481}{\deci\meter}
$$
\item \begin{enumerate}[label=\textbf{\xslalph*})]
\setcounter{enumii}{1}
\item Za koliko se mora povečati stranica kocke, da se bo njena prostornina s tem povečala za četrtino (prejšnje vrednosti)?
$$
\left(n+x\right)^2=x^2\cdot\frac{5}{4} \rightarrow n+x=\frac{\sqrt{5}x}{2} \rightarrow n=\frac{\sqrt{5}x}{2}-x
$$
\end{enumerate}
\item \begin{enumerate}[label=\textbf{\xslalph*})]
\setcounter{enumii}{1}
\item V veliki steklenici je 10 litrov vina. Iz nje izlijemo $x$ litrov vina in nato dolijemo $x$ litrov vode. Zatem ponovno izlijemo iz steklenice $x$ litrov te mešanice in za \say{nadomestek} spet dolijemo $x$ litrov vode. In zatem še enkrat tako. Tedaj je v sodu 5 litrov vina in 5 litrov vode. Izračuaj $x$.
\def\de{\left(1-x\right)-x\left(\frac{1-x}{1}\right)}
$$
\left(\de\right)-x\frac{\de}{1}=\frac{1}{2}=1-x-x\left(1-x\right)-x\left(1-x-x\left(1-x\right)\right)=-x^3+3x^2-3x+1=
$$
$$
=-\left(x^3-1\right)+3x\left(x-1\right)=-\left(x-1\right)\cdot\left(x^2+x+1\right)+3x\left(x-1\right)
=-\left(x-1\right)\cdot\left(x^2+x+1-3x\right)=
$$
$$
-\left(x-1\right)\cdot\left(x^2-2x+1\right)=-\left(x-1\right)\cdot\left(x-1\right)^2=-\left(x-1\right)^3=\frac{1}{2}
\rightarrow \left(x-1\right)^3=\frac{-1}{2} \rightarrow x=-\frac{\sqrt[3]{4}}{2}+1
$$
Sedaj pa $x$ za pravilno vrednost še pomnožimo z 10 in dobimo 2,05299, torej je, ker smo prej namesto \SI{10}{\liter} uporabili 1, rezultat \SI{2,05299}{\liter}.
\end{enumerate}
\end{enumerate}
\section{Zaključek}
Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in
\hologo{LaTeX}\footnote{Za izdelavo dokumenta potrebujete \texttt{TeXLive 2020}.}
izvorna koda, zgodovina sprememb in prejšnje različice, je na voljo v mojem šolskem Git repozitoriju na
\url{https://git.sijanec.eu/sijanec/sola-gimb-2} v mapi
\href{https://git.sijanec/sola-gimb-2/src/branch/master/\predmdn/\predmkaj/\stevilkadn/}{/\predmdn/\predmkaj/\stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/\predmdn/\predmkaj/\stevilkadn/dokument.pdf} in/ali \url{https://git.sijanec.eu/sijanec/sola-gimb-2/raw/branch/master/\predmdn/\predmkaj/\stevilkadn/dokument.pdf}.
\if\razhroscevanje1
\vfill
\section*{Razhroščevalne informacije}
Te informacije so generirane, ker je omogočeno razhroščevanje. Pred objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroscevanje} na 0 v začetku dokumenta.
Grafi imajo natančnost \functionSamples\space točk na graf.
Konec generiranja dokumenta: \today\ ob \currenttime\footnote{To ne nakazuje dejanskega časa, ko je bil dokument napisan, temveč čas, ko je bi dokument generiran v PDF/DVI obliko. Isto velja za datum v glavi dokumenta. Če berete direktno iz LaTeX datoteke, bo to vedno današnji datum.}%\input|"date -Ins"
Dokument se je generiral R0qK1KR2 \SI{}{\second}.
\fi
% \item $$$$ aaasecgeninsaaa R0qK1KR2
\end{document}
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