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Quadratwurzeln vereinfachen, Nenner rational machen

Deutsche Übersetzung von https://numbas.mathcentre.ac.uk/question/22587/using-surds-rationalising-the-denominator/ von Elliott Fletcher

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Benutzen Sie sqrt(), um eine Quadratwurzel einzugeben. Um zum Beispiel $\\sqrt{3}$ einzugeben, tippen Sie sqrt(3) in dem Eingabefeld. Um zum Beispiel $3\\sqrt{5}$ einzugeben, tippen Sie 3*sqrt(5) in das Eingabefeld.

", "advice": "

a)

Für nicht-negative reelle Zahlen $a$, $b$ gilt

\\[\\sqrt{ab} = \\sqrt{a} \\cdot \\sqrt{b}.\\]

Ein Wurzelausdruck kann vereinfacht werden, wenn die Zahl unter der Wurzel als Produkt einer Quadratzahl $>1$ und einer weiteren Zahl geschrieben werden kann.

In unserem Fall ist $\\var{p}$ eine Primzahl.

Folglich kann $\\sqrt{\\var{p}}$ nicht weiter vereinfacht werden.

Genauso ist $\\var{a}$ eine Primzahl, und $\\sqrt{\\var{a}}$ kann ebenfalls nicht weiter vereinfacht werden.

Andererseits können wir $\\simplify{{a}*{n}^2}$ als Produkt schreiben und $\\sqrt{\\simplify{{a}*{n}^2}}$ vereinfachen zu

\\[
\\begin{align}
\\sqrt{\\simplify{{a}*{n}^2}} &= \\sqrt{\\simplify{{n}^2}} \\cdot \\sqrt{\\var{a}}\\\\
&= \\simplify{{n}*sqrt({a})}.
\\end{align}
\\]

b)

Mit demselben Wurzelgesetz wie in Teil a), lässt sich $\\sqrt{\\simplify{{n}^2*{p}}}$ vereinfachen zu

\\[
\\begin{align}
\\sqrt{\\simplify{{n}^2*{p}}} &= \\sqrt{\\simplify{{n}^2}} \\cdot \\sqrt{\\var{p}}\\\\
&= \\simplify{{n}*sqrt({p})}.
\\end{align}
\\]

c)

In diesem Abschnitt benutzen wir die folgenden Regeln (für nicht-negative reelle Zahlen $a$, $b$; für die zweite Gleichheit muss $b\\ne 0$ gelten).

\\[\\sqrt{ab} = \\sqrt{a} \\cdot \\sqrt{b} \\text{.} \\]

\\[ \\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}} \\text{.} \\]

Damit vereinfachen wir $\\displaystyle\\frac{ \\sqrt{\\simplify{{a}*{v}}} }{ \\sqrt{\\var{a}} }$ wie folgt:

\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\frac{\\sqrt{\\var{a}} \\cdot \\sqrt{\\var{v}}}{\\sqrt{\\var{a}}} \\\\[0.5em]
&= \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\cdot \\sqrt{\\var{v}} \\\\[0.5em]
&= \\simplify{{sqrt(a)/sqrt(a)}} \\cdot \\sqrt{\\var{v}} \\\\[0.5em]
&= \\sqrt{\\var{v}} \\text{.}
\\end{align}
\\]

Oder

\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\sqrt{\\frac{\\simplify{{a}*{v}}}{\\var{a}}} \\\\[0.5em]
&= \\sqrt{\\var{v}} \\text{.}
\\end{align}
\\]

d)

Wir vereinfachen den Bruch folgendermaßen

\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}} &= \\frac{\\sqrt{\\simplify{({b*m})^2}} \\cdot \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em]
&= \\frac{\\simplify{{b*m}} \\cdot \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em]
&= \\simplify{{b}*sqrt({s})} \\text{.}
\\end{align}
\\]

e)

\\[
\\begin{align}
\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2{a})+{n}sqrt({b}^2*{a})} &= \\var{d}\\sqrt{\\var{a}} - \\var{b}(\\sqrt{\\simplify{{v}^2}} \\cdot \\sqrt{\\var{a}})+\\var{n}(\\sqrt{\\simplify{{b}^2}} \\cdot \\sqrt{\\var{a}}) \\\\
&= \\var{d}\\sqrt{\\var{a}} -\\var{b}(\\simplify{{v}*sqrt({a})})+\\var{n}(\\simplify{{b}*sqrt({a})}) \\\\
&= \\simplify{{d}sqrt({a})}-\\simplify{{b}*{v}sqrt({a})}+\\simplify{{n}*{b}sqrt({a})} \\\\
&= \\simplify{({d}-{b}*{v}+{n}*{b})sqrt({a})} \\text{.}
\\end{align}
\\]

f)

Wir schreiben $\\displaystyle\\frac{1}{\\sqrt{a}}$ als Bruch mit einer ganzen Zahl im Nenner, indem wir mit $\\sqrt{a}$ erweitern.

\\[
\\begin{align}
\\frac{1}{\\sqrt{\\var{a}}} &= \\frac{1}{\\sqrt{\\var{a}}} \\cdot \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\\\[0.5em]
&= \\frac{\\sqrt{\\var{a}}}{\\var{a}} \\text{.}
\\end{align}
\\]

g)

In diesem Fall erweitern wir mit $a-\\sqrt{b}$.

\\[
\\begin{align}
\\frac{1}{\\var{n}+\\sqrt{\\var{a}}} &= \\frac{1}{\\var{n}+\\sqrt{\\var{a}}} \\cdot \\frac{\\var{n}-\\sqrt{\\var{a}}}{\\var{n}-\\sqrt{\\var{a}}} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{(\\var{n}+\\sqrt{\\var{a}})(\\var{n}-\\sqrt{\\var{a}})} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2}-\\var{a}} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2-{a}}} \\text{.}
\\end{align}
\\]

h)

Wir erweitern mit $a+\\sqrt{b}$.

\\[
\\begin{align}
\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} &= \\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} \\cdot \\frac{\\var{d+p}+\\sqrt{\\var{p}}}{\\var{d+p}+\\sqrt{\\var{p}}} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{(\\var{d+p}-\\sqrt{\\var{p}})(\\var{d+p}+\\sqrt{\\var{p}})} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2}-\\var{p}} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2-{p}}} \\\\[0.5em]
&=\\simplify{{t}/{(d+p)^2-p}}(\\var{d+p}+\\sqrt{\\var{p}}) \\\\[0.5em]
&= \\simplify[all,!noleadingMinus]{({t*(d+p)}+{t}*sqrt({p}))/({(d+p)^2-p})} \\text{.}
\\end{align}
\\]

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all numbers from 3-10 for parts a, b, e, g

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Parts c and e

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Fraction in answer for part d.

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parts b and d

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Parts a, d,e and h

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Short list of primes for part d.

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parts d and e

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Welche der folgenden Ausdrücke lassen sich weiter vereinfachen?

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Kann vereinfacht werden

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Kann nicht weiter vereinfacht werden

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$\\sqrt{\\var{p}}$

", "

$\\sqrt{\\simplify{{a}*{n}^2}}$

", "

$\\sqrt{\\var{a}}$

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Vereinfachen Sie $\\sqrt{\\simplify{{n}^2*{p}}}$.

$\\sqrt{\\simplify{{n}^2*{p}}} =$ [[0]]$\\sqrt{\\var{p}}$.

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Es gilt (für nicht-negative reelle Zahlen $a$, $b$)

$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}$.

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Vereinfachen Sie $\\displaystyle\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}}$.

$\\displaystyle\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} =$ [[0]].

Sie können eine der folgenden Tatsachen benutzen (für nicht-negative reelle Zahlen $a,b$):

$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}$.

$\\displaystyle\\sqrt{\\frac{a}{b}} = \\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}$.

You must simplify your answer further.

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You must simplify your answer further.

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Vereinfachen Sie $\\displaystyle\\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}}$.

$\\displaystyle\\frac{\\sqrt{\\simplify{({b}*{m})^2*{s}}}}{\\var{m}} =$ [[0]]$\\sqrt{\\var{s}}$.

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Vereinfachen Sie $\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2*{a})+{n}sqrt({b}^2*{a})}$.

$\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2*{a})+{n}sqrt({b}^2*{a})} =$ [[0]].

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Schreiben Sie $\\displaystyle\\frac{1}{\\sqrt{\\var{a}}}$ als Bruch mit einer ganzen Zahl im Nenner.

$\\displaystyle\\frac{1}{\\sqrt{\\var{a}}} =$ [[0]] [[1]] .

Um die Wurzel im Nenner von $\\frac{1}{\\sqrt{a}}$ \"loszuwerden\", erweitern Sie mit $\\sqrt{a}$.

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Schreiben Sie $\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}}$ als Bruch mit einer ganzen Zahl im Nenner.

$\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}} =$ [[0]] [[1]] .

In diesem Fall können Sie den Wurzelausdruck im Nenner von $\\displaystyle\\frac{1}{a+\\sqrt{b}}$ loswerden, indem Sie mit $a-\\sqrt{b}$ erweitern (und die dritte binomische Formel benutzen).

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Schreiben Sie $\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}}$ als Bruchzahl mit einer ganzen Zahl als Nenner.

$\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} =$ [[0]] [[1]] .

Erweitern Sie mit ${a+\\sqrt{b}}$.

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