Benutzer:NickK~dewikibooks/ Seminaraufgaben

Seminar 1

Faktorisieren

a) $(x+y)^{2}-x-y+(2x+2y)\cdot 3x\,$

b) $6fk-15fl+3f+2gk-5gl+g-4hk+10hl-2h\,$

c) $-e^{2x}\cdot (-x^{2}+3x-7)\cdot (-2)+(x^{2}-2x+14)\cdot e^{x}$

Ausmultiplizieren

a) $-2xy({\frac {15}{4}}x^{2}y)(-x+{\frac {2}{3}}xy-{\frac {1}{3}}y)$

b) $(2-5x)(3+7x)(10x+4)\,$ 

c) $(f^{2}-{\frac {1}{3}}g^{2})(-3e^{2})-e(-{\frac {3}{2}}ef+eg)(2f-g)$

d) $\lbrack -3+(-{\frac {9}{2}}a+5)\rbrack \cdot \lbrack 2a-(3-{\frac {1}{2}}a)\rbrack$

Bruchterme zusammenfassen

a) ${\frac {11a-3}{2x+3}}-{\frac {7a-4}{2x+2}}+{\frac {5a-6}{6x+6}}$

b) ${\frac {{\frac {1}{s^{2}-1}}-{\frac {1}{s^{2}}}}{2+{\frac {1}{s-1}}-{\frac {1}{s+1}}}}$
 
 
c) $1-{\frac {u}{1-{\frac {u}{u+1}}}}$

Potenzen

a) ${\frac {4x^{2-m}\cdot y^{3m}}{7z^{m-n}}}\div {\frac {5z^{m+n}\cdot x^{3-m}}{14y^{1-2m}}}$

b) ${\frac {12{\frac {1}{x^{2}}}y^{3}}{8z^{2}}}\cdot {\frac {4{\frac {1}{y^{2}}}z}{3{\frac {1}{x^{5}}}}}\div {\frac {6{\frac {1}{z^{3}}}}{2{\frac {1}{y^{4}}}z}}$

Binomische Formeln

a) $\left(3xy^{2}-{\frac {4}{5}}x^{3}y^{7}\right)^{2}$

Wandle um!
c) $\!a^{2}-b\,$

d) ${\frac {1}{49}}s^{4}t^{2}+4s^{3}t^{5}+196s^{2}t^{8}$

Quadratische Ergänzung!
f) $\!3x^{2}-9x+5\,$

g) $\!2x^{2}-12x=32\,$

Wurzeln

a) ${\sqrt {0,0121}}\quad \quad {\sqrt {6,25\cdot 10^{6}}}\quad \quad {\sqrt[{3}]{8,1\cdot 10^{-11}}}$

b) Teilradizieren:
 ${\sqrt {72a^{2}b}}\quad \quad {\frac {{\sqrt {8}}-{\sqrt {2}}}{{\sqrt {8}}+{\sqrt {2}}}}$

c) Faktoren in die Wurzel hineinziehen:

   $3\cdot {\sqrt {7}}\quad \quad 4\cdot {\sqrt[{3}]{18}}\quad \quad x\cdot {\sqrt {1-{\frac {y^{2}}{x^{2}}}}}$

d) ${\sqrt[{3}]{0,216}}\quad \quad {\sqrt[{3}]{81a^{5}b^{7}}}\div {\sqrt[{3}]{3ab}}$

e) $\left({\sqrt {8}}\right)^{\frac {2}{3}}\cdot \left({\sqrt {4}}\right)^{\frac {5}{2}}$

f) $\left({\sqrt {\frac {a}{b}}}+{\sqrt {\frac {b}{a}}}\right)^{2}$

g) ${\sqrt {5}}\cdot {\frac {{\sqrt {\frac {3}{10}}}+{\sqrt {\frac {10}{3}}}}{{\sqrt {\frac {3}{2}}}-{\sqrt {\frac {2}{3}}}}}$

Logarithmen

Ohne Taschenrechner!!
a)$\log _{2}\left({\frac {1}{8}}\right)\quad \quad \log _{8}4\quad \quad \log _{\sqrt {2}}\left({\frac {1}{2}}\right)$

Umformen!

b)$\log _{a}\left({\frac {a^{7}-a^{4}}{b}}\right)$

Gleichungen

a) $\!13x^{2}-9x^{4}=4\,$

b) ${\frac {32}{x-1}}-{\frac {45}{x-2}}=1$

c) ${\frac {x-1}{x-2}}-{\frac {9}{x+1}}={\frac {3}{x^{2}-x-2}}$

d) $2{\sqrt {2x+9}}+{\sqrt {5-x}}={\sqrt {29+x}}$

e) $\!x^{3}+3x^{2}-78x-80=0\,$

f) $3^{2x}-2\cdot 3^{x+1}-7=0$

g) $\!\log _{5}{x}+2\ln {x}=2+2\ln {25}\,$

Ungleichungen/Betragsgleichungen

a) ${\frac {4x+3}{{\frac {5}{2}}-x}}\leq 6$

b) $\!|3-2x|>5\,$

c) ${\frac {3x+2}{|x+5|}}\geq 1$