a) ( x + y ) 2 − x − y + ( 2 x + 2 y ) ⋅ 3 x {\displaystyle (x+y)^{2}-x-y+(2x+2y)\cdot 3x\,} b) 6 f k − 15 f l + 3 f + 2 g k − 5 g l + g − 4 h k + 10 h l − 2 h {\displaystyle 6fk-15fl+3f+2gk-5gl+g-4hk+10hl-2h\,} c) − e 2 x ⋅ ( − x 2 + 3 x − 7 ) ⋅ ( − 2 ) + ( x 2 − 2 x + 14 ) ⋅ e x {\displaystyle -e^{2x}\cdot (-x^{2}+3x-7)\cdot (-2)+(x^{2}-2x+14)\cdot e^{x}}
a) − 2 x y ( 15 4 x 2 y ) ( − x + 2 3 x y − 1 3 y ) {\displaystyle -2xy({\frac {15}{4}}x^{2}y)(-x+{\frac {2}{3}}xy-{\frac {1}{3}}y)} b) ( 2 − 5 x ) ( 3 + 7 x ) ( 10 x + 4 ) {\displaystyle (2-5x)(3+7x)(10x+4)\,} c) ( f 2 − 1 3 g 2 ) ( − 3 e 2 ) − e ( − 3 2 e f + e g ) ( 2 f − g ) {\displaystyle (f^{2}-{\frac {1}{3}}g^{2})(-3e^{2})-e(-{\frac {3}{2}}ef+eg)(2f-g)} d) [ − 3 + ( − 9 2 a + 5 ) ] ⋅ [ 2 a − ( 3 − 1 2 a ) ] {\displaystyle \lbrack -3+(-{\frac {9}{2}}a+5)\rbrack \cdot \lbrack 2a-(3-{\frac {1}{2}}a)\rbrack }
a) 11 a − 3 2 x + 3 − 7 a − 4 2 x + 2 + 5 a − 6 6 x + 6 {\displaystyle {\frac {11a-3}{2x+3}}-{\frac {7a-4}{2x+2}}+{\frac {5a-6}{6x+6}}} b) 1 s 2 − 1 − 1 s 2 2 + 1 s − 1 − 1 s + 1 {\displaystyle {\frac {{\frac {1}{s^{2}-1}}-{\frac {1}{s^{2}}}}{2+{\frac {1}{s-1}}-{\frac {1}{s+1}}}}} c) 1 − u 1 − u u + 1 {\displaystyle 1-{\frac {u}{1-{\frac {u}{u+1}}}}}
a) 4 x 2 − m ⋅ y 3 m 7 z m − n ÷ 5 z m + n ⋅ x 3 − m 14 y 1 − 2 m {\displaystyle {\frac {4x^{2-m}\cdot y^{3m}}{7z^{m-n}}}\div {\frac {5z^{m+n}\cdot x^{3-m}}{14y^{1-2m}}}} b) 12 1 x 2 y 3 8 z 2 ⋅ 4 1 y 2 z 3 1 x 5 ÷ 6 1 z 3 2 1 y 4 z {\displaystyle {\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}}}
a) ( 3 x y 2 − 4 5 x 3 y 7 ) 2 {\displaystyle \left(3xy^{2}-{\frac {4}{5}}x^{3}y^{7}\right)^{2}} Wandle um! c) a 2 − b {\displaystyle \!a^{2}-b\,} d) 1 49 s 4 t 2 + 4 s 3 t 5 + 196 s 2 t 8 {\displaystyle {\frac {1}{49}}s^{4}t^{2}+4s^{3}t^{5}+196s^{2}t^{8}} Quadratische Ergänzung! f) 3 x 2 − 9 x + 5 {\displaystyle \!3x^{2}-9x+5\,} g) 2 x 2 − 12 x = 32 {\displaystyle \!2x^{2}-12x=32\,}
a) 0 , 0121 6 , 25 ⋅ 10 6 8 , 1 ⋅ 10 − 11 3 {\displaystyle {\sqrt {0,0121}}\quad \quad {\sqrt {6,25\cdot 10^{6}}}\quad \quad {\sqrt[{3}]{8,1\cdot 10^{-11}}}} b) Teilradizieren: 72 a 2 b 8 − 2 8 + 2 {\displaystyle {\sqrt {72a^{2}b}}\quad \quad {\frac {{\sqrt {8}}-{\sqrt {2}}}{{\sqrt {8}}+{\sqrt {2}}}}} c) Faktoren in die Wurzel hineinziehen: 3 ⋅ 7 4 ⋅ 18 3 x ⋅ 1 − y 2 x 2 {\displaystyle 3\cdot {\sqrt {7}}\quad \quad 4\cdot {\sqrt[{3}]{18}}\quad \quad x\cdot {\sqrt {1-{\frac {y^{2}}{x^{2}}}}}} d) 0 , 216 3 81 a 5 b 7 3 ÷ 3 a b 3 {\displaystyle {\sqrt[{3}]{0,216}}\quad \quad {\sqrt[{3}]{81a^{5}b^{7}}}\div {\sqrt[{3}]{3ab}}} e) ( 8 ) 2 3 ⋅ ( 4 ) 5 2 {\displaystyle \left({\sqrt {8}}\right)^{\frac {2}{3}}\cdot \left({\sqrt {4}}\right)^{\frac {5}{2}}} f) ( a b + b a ) 2 {\displaystyle \left({\sqrt {\frac {a}{b}}}+{\sqrt {\frac {b}{a}}}\right)^{2}} g) 5 ⋅ 3 10 + 10 3 3 2 − 2 3 {\displaystyle {\sqrt {5}}\cdot {\frac {{\sqrt {\frac {3}{10}}}+{\sqrt {\frac {10}{3}}}}{{\sqrt {\frac {3}{2}}}-{\sqrt {\frac {2}{3}}}}}}
Ohne Taschenrechner!! a) log 2 ( 1 8 ) log 8 4 log 2 ( 1 2 ) {\displaystyle \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 ( a 7 − a 4 b ) {\displaystyle \log _{a}\left({\frac {a^{7}-a^{4}}{b}}\right)}
a) 13 x 2 − 9 x 4 = 4 {\displaystyle \!13x^{2}-9x^{4}=4\,} b) 32 x − 1 − 45 x − 2 = 1 {\displaystyle {\frac {32}{x-1}}-{\frac {45}{x-2}}=1} c) x − 1 x − 2 − 9 x + 1 = 3 x 2 − x − 2 {\displaystyle {\frac {x-1}{x-2}}-{\frac {9}{x+1}}={\frac {3}{x^{2}-x-2}}} d) 2 2 x + 9 + 5 − x = 29 + x {\displaystyle 2{\sqrt {2x+9}}+{\sqrt {5-x}}={\sqrt {29+x}}} e) x 3 + 3 x 2 − 78 x − 80 = 0 {\displaystyle \!x^{3}+3x^{2}-78x-80=0\,} f) 3 2 x − 2 ⋅ 3 x + 1 − 7 = 0 {\displaystyle 3^{2x}-2\cdot 3^{x+1}-7=0} g) log 5 x + 2 ln x = 2 + 2 ln 25 {\displaystyle \!\log _{5}{x}+2\ln {x}=2+2\ln {25}\,}
a) 4 x + 3 5 2 − x ≤ 6 {\displaystyle {\frac {4x+3}{{\frac {5}{2}}-x}}\leq 6} b) | 3 − 2 x | > 5 {\displaystyle \!|3-2x|>5\,} c) 3 x + 2 | x + 5 | ≥ 1 {\displaystyle {\frac {3x+2}{|x+5|}}\geq 1}
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Wandle um!
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Quadratische Ergänzung!
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![{\displaystyle {\sqrt {0,0121}}\quad \quad {\sqrt {6,25\cdot 10^{6}}}\quad \quad {\sqrt[{3}]{8,1\cdot 10^{-11}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87287a90ffae6927b490fb6c74e33ee7412dd3b4)
b) Teilradizieren:
c) Faktoren in die Wurzel hineinziehen:
![{\displaystyle 3\cdot {\sqrt {7}}\quad \quad 4\cdot {\sqrt[{3}]{18}}\quad \quad x\cdot {\sqrt {1-{\frac {y^{2}}{x^{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ee0d1aaf00a24850070a2e38d561a26223c264)
d) ![{\displaystyle {\sqrt[{3}]{0,216}}\quad \quad {\sqrt[{3}]{81a^{5}b^{7}}}\div {\sqrt[{3}]{3ab}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00f46ffa0b9202f341ebbb3f08aee5c0743dfc4c)
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Umformen!
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