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11. Hausaufgabe

• 11. Hausaufgabe: Algebra Buch, Seite 34, Aufgabe 10o:
  Multipliziere und radiziere so weit wie möglich:
  $\left(1+\sqrt{2}+\sqrt{3}\right)^2=1+\sqrt{2}+\sqrt{3}+\sqrt{2}+2+\sqrt{6}+\sq... ...2}+2\cdot{}\sqrt{3}+2\sqrt{6}=6+2\cdot{}\left(\sqrt{2}+\sqrt{3}+\sqrt{6}\right)$
• Hausaufgabe: Algebra Buch, Seite 33, Aufgabe 4:
  Radiziere teilweise!

  e)

  $5\cdot{}\sqrt{192}=5\cdot{}\sqrt{64\cdot{}3}=5\cdot{}8\cdot{}\sqrt{3}=40\cdot{}\sqrt{3}$

  f)

  $3\cdot{}\sqrt{288}=3\cdot{}\sqrt{144\cdot{}2}=3\cdot{}12\cdot{}\sqrt{2}=36\cdot{}\sqrt{2}$

  g)

  $2\cdot{}\sqrt{507}=2\cdot{}\sqrt{169\cdot{}3}=2\cdot{}13\cdot{}\sqrt{3}=26\cdot{}\sqrt{3}$

  h)

  $5\cdot{}\sqrt{980}=5\cdot{}\sqrt{196\cdot{}5}=5\cdot{}14\cdot{}\sqrt{5}=70\cdot{}\sqrt{5}$

  i)

  $\sqrt{ab^2}=\sqrt{a}\cdot{}\sqrt{b^2}=\sqrt{a}\cdot{}\left\vert b\right\vert=\... ...dge a \geq 0 \\ -\sqrt{a}\cdot{}b&\text{falls }b<0 \wedge a \geq 0 \end{cases}$

  k)

  $\sqrt{25r^2s}=5\cdot{}\left\vert r\right\vert\cdot{}\sqrt{s}=\begin{cases} \te... ...eq 0 \\ 5\cdot{}r\cdot{}-\sqrt{s}&\text{falls }r<0 \wedge s \geq 0 \end{cases}$

  l)

  $\sqrt{81a^4b^2c}=9\cdot{}a^2\cdot{}\left\vert b\right\vert\cdot{}\sqrt{c}=\beg... ...\wedge c \geq 0 \\ -9a^2b\sqrt{c}&\text{falls }b<0 \wedge c \geq 0 \end{cases}$

  m)

  $\sqrt{x^3}=\sqrt{x^2\cdot{}x}=\left\vert x\right\vert\cdot{}\sqrt{x}=\begin{ca... ...sqrt{x}&\text{falls }x<0 \text{Nicht definiert} & \text{falls } x<0 \end{cases}$

  n)

  $\sqrt{27p^3q^2r}=\sqrt{3^2\cdot{}3\cdot{}p^2\cdot{}p\cdot{}q^2\cdot{}r}=3\cdot... ...\vee \left( p \geq 0 \wedge q < 0 \right) \wedge p \cdot{} r \geq 0 \end{cases}$

  o)

  $\sqrt{x^2+y^2}=\begin{cases} \text{Nicht definiert} & \text{falls } x^2+y^2 < 0 \\ \sqrt{x^2+y^2} & \text{falls } x^2+y^2 \geq 0 \end{cases}$

  p)

  $\sqrt{a^2x+a^2y}=\sqrt{a^2\cdot{}(x+y)}=\left\vert a\right\vert\cdot{}\sqrt{x+... ...x+y \geq 0 \\ -a\sqrt{x+y} & \text{falls } a < 0 \wedge x+y \geq 0 \end{cases}$

• Hausaufgabe: Algebra Buch, Seite 33, Aufgabe 5l:
  Ziehe unter das Wurzelzeichen!
  $\frac{xy}{4z}\cdot{}\sqrt{\frac{16z^3}{x^2y}}=\sqrt{\frac{16z^3}{x^2y}\cdot{}\... ...left(z<0 \wedge y<0\right) \vee \left(z\geq0 \wedge y\geq0\right)}} \end{cases}$
• Hausaufgabe: Algebra Buch, Seite 33, Aufgabe 6:
  Berechne:

  a)

  $\sqrt{9}+\sqrt{16}-\sqrt{9+16}=3+4-\sqrt{25}=3+4-5=2$

  b)

  $\sqrt{9}\cdot{}\sqrt{16}-\sqrt{9\cdot{}16}=\sqrt{9\cdot{}16}-\sqrt{9\cdot{}16}=0$

  c)

  $\sqrt{169}-\sqrt{25}-\sqrt{169-25}=13-5-\sqrt{144}=13-5-12=-4$

  d)

  $\frac{\sqrt{169}:\sqrt{25}}{\sqrt{169:25}}=\frac{\sqrt{169:25}}{\sqrt{169:25}}=1$

  e)

  $\left(\sqrt{144}\cdot{}\sqrt{25}-\sqrt{144\cdot{}25}\right)^2=\left(\sqrt{144\cdot{}25}-\sqrt{144\cdot{}25}\right)=0$

  f)

  $\left(\sqrt{144}+\sqrt{25}-\sqrt{144+25}\right)^2=\left(12+5-13\right)^2=\left(-4\right)^2=16$

  g)

  $\sqrt{1\frac{9}{16}}-\sqrt{1\frac{9}{16}}+\sqrt{1\cdot{}\frac{9}{16}}=\sqrt{9}{16}=\frac{3}{4}$

  h)

  $\sqrt{9\cdot{}\frac{25}{16}}-\sqrt{10\frac{9}{16}}+\sqrt{9\frac{25}{16}}=3\cdot{}\frac{5}{4}-\sqrt{\frac{169}{16}}=\frac{13}{4}=3\frac{1}{4}$

• Hausaufgabe: Algebra Buch, Seite 34, Aufgabe 7:
  Fasse so weit wie möglich zusammen!

  a)

  $5\cdot{}\sqrt{2}+3\cdot{}\sqrt{2}=\sqrt{2}\cdot{}\left(5+3\right)=\sqrt{2}\cdot{}8$

  b)

  $\sqrt{7}+\sqrt{7}=2\cdot{}\sqrt{7}$

  c)

  $\sqrt{3}-\sqrt{2}=\sqrt{3}-\sqrt{2}$

  d)

  $3\cdot{}\sqrt{2}-2\cdot{}\sqrt{3}+\sqrt{2}=\sqrt{3}\cdot{}\left(3-2\right)+\sqrt{2}=\sqrt{3}+\sqrt{2}$

  e)

  $7\cdot{}\sqrt{3}-3\cdot{}\sqrt{2}+\sqrt{3}=8\cdot{}\sqrt{3}-3\cdot{}\sqrt{2}=\sqrt{8^2\cdot{}3}-\sqrt{3^2\cdot{}2}=\sqrt{192}-\sqrt{18}$

  f)

  $\sqrt{5}-4\cdot{}\sqrt{7}+4\cdot{}\sqrt{5}+\sqrt{7}=5\cdot{}\sqrt{5}-3\cdot{}\sqrt{7}=\sqrt{5^2\cdot{}5}-\sqrt{3^2\cdot{}7}=\sqrt{625}-\sqrt{63}=25-\sqrt{63}$

  g)

  $2\cdot{}\sqrt{5}-6\cdot{}\sqrt{2}+3\cdot{}\sqrt{5}+\sqrt{2}=5\cdot{}\sqrt{5}-5\cdot{}\sqrt{2}=5\cdot{}\left(\sqrt{5}-\sqrt{2}\right)$

  h)

  $5\cdot{}\sqrt{a}-4\cdot{}\sqrt{b}+\sqrt{a}-\sqrt{b}=6\cdot{}\sqrt{a}-5\cdot{}\... ... 0 \\ 6\sqrt{a}-5\sqrt{b} & \text{falls } a \geq 0 \wedge b \geq 0 \end{cases}$

  i)

  $\sqrt{a}+\sqrt{b}-\sqrt{a+b}= \begin{cases} \text{Nicht definiert} & \text{fal... ...a}+\sqrt{b}-\sqrt{a+b} & \text{falls } a \geq 0 \wedge b \geq 0 \\ \end{cases}$

  k)

  $2\cdot{}\sqrt{a}+\sqrt{b}-3\cdot{}\sqrt{a+b}-3\cdot{}\sqrt{a}+3\cdot{}\sqrt{b}... ...t{a}+4\sqrt{b}-3\sqrt{a+b} & \text{falls } a \geq 0 \wedge b \geq 0 \end{cases}$

• Hausaufgabe: Algebra Buch, Seite 34, Aufgabe 8:
  Fasse so weit wie möglich zusammen!

  a)

  $\sqrt{45}+\sqrt{20}+\sqrt{5}=\sqrt{5\cdot{}9}+\sqrt{4\cdot{}5}+\sqrt{5}=\sqrt{5}\cdot{}3+2\cdot{}\sqrt{5}+\sqrt{5}=6\cdot{}\sqrt{5}$

  b)

  $\sqrt{72}-\sqrt{18}+\sqrt{8}-\sqrt{2}=\sqrt{2\cdot{}36}-\sqrt{2\cdot{}9}+\sqrt... ...}=\sqrt{2}\cdot{}6-\sqrt{2}\cdot{}3+\sqrt{2}\cdot{}2-\sqrt{2}=12\cdot{}\sqrt{2}$

  c)

  $\sqrt{2}-\sqrt{4}+\sqrt{8}-\sqrt{16}+\sqrt{32}-\sqrt{64}=\sqrt{2^1}-\sqrt{2^2}... ...-2^3=1\cdot{}\sqrt{2}+2\cdot{}\sqrt{2}-4+4\cdot{}\sqrt{2}-8=7\cdot{}\sqrt{2}-12$

  d)

  $\sqrt{3}-\sqrt{9}+\sqrt{27}-\sqrt{81}+\sqrt{243}-\sqrt{729}=\sqrt{3}+\sqrt{3^2... ...3}-9+9\cdot{}\sqrt{3}-27=13\cdot{}\sqrt{3}-36=13\cdot{}\left(\sqrt{3}-12\right)$

  e)

  $\sqrt{75}-2\cdot{}\sqrt{3}-2\cdot{}\sqrt{27}=\sqrt{25\cdot{}3}-\sqrt{4\cdot{}3... ...sqrt{2}-3\cdot{}2\cdot{}\sqrt{3}=3\cdot{}\left(\sqrt{2}-2\cdot{}\sqrt{3}\right)$

  f)

  $\sqrt{242}-\sqrt{98}+4\cdot{}\sqrt{8}-\frac{2}{5}\cdot{}\sqrt{50}-\sqrt{200}=\... ...\sqrt{2}=4\cdot{}\sqrt{2}+8\cdot{}\sqrt{2}-2\cdot{}\sqrt{2}-10\cdot{}\sqrt{2}=0$

  g)

  $2\cdot{}\sqrt{45}-4\cdot{}\sqrt{63}+\sqrt{245}+\sqrt{28}=\sqrt{2^2\cdot{}45}-\... ...rt{5}-12\cdot{}\sqrt{7}+7\cdot{}\sqrt{5}+2\cdot{}\sqrt{7}=13\sqrt{5}-10\sqrt{7}$

  h)

  $\frac{7}{3}\sqrt{999}+\frac{2}{5}\sqrt{275}+\frac{3}{4}\sqrt{176}=\sqrt{\frac{... ...t{111}+2\cdot{}\sqrt{11}+3\cdot{}\sqrt{11}=7\cdot{}\sqrt{111}+5\cdot{}\sqrt{11}$

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