TitelPotenzrechenregeln
Autoranonym
veröffentlicht05.06.2023
FachMathematik

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Potenzrechenregeln

Potenz

Für $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} n\in\mathbb{N}$ erhält man $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^n$ indem man $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a$ n-mal mit sich selbst multipliziert:

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^n=\underbrace{a\cdot\dots\cdot a}_{n-mal}$

Bsp.:

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}$ 1) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^3=2\cdot2\cdot2=8$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}$ 2) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3^2=3\cdot3$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}$ 3) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-3)^2=(-3)\cdot(-3)=9$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}$ 4) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 0{,}2^4=0{,}0016$

Potenzen binden stärker als Rechen- und Vorzeichen:

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}$ 5) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} -3^2=-3\cdot 3=-9$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}$ 6) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3\cdot4^2=3\cdot 16=48$

Berechnen Sie im Kopf! Vereinfachen Sie so weit wie möglich!

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^2=\cloze{16}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 8^2=\cloze{64}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^4=\cloze{256}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-5)^2=\cloze{25}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} -5^2=\cloze{-25}$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-5)^3=\cloze{\small-125}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 0{,}5^3=\cloze{\small0{,}125}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 50^3=\cloze{\small125000}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{2}{5})^3=\cloze{\small\textstyle\frac{2}{5}\cdot\frac{2}{5}\cdot\frac{2}{5}=\frac{8}{125}}$

Tipp: Nutzen Sie eine Merkhilfe, wenn Sie sich nicht sicher sind:

Für

\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a, b\in\mathbb{R}

und

\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} n, m\in\mathbb{N}

gelten bestimmte Potenzrechenregeln. Ergänzen Sie diese!

(A) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^n\cdot a^m=\cloze{a^{n+m}}$
(B) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^n\cdot b^n=\cloze{(a\cdot b)^n}$
C) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (a^n)^m=\cloze{a^{n\cdot m}}$
D) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^{-n}=\cloze{\textstyle\frac{1}{a^n}}$
E) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^{\frac{1}{n}}=\cloze{\sqrt[n]{a}}$

Notieren Sie jeweils auf dem Gleichheitszeichen, welche Regel Sie anwenden und ergänzen Sie die Beispiele!

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3^{-2}\cloze{\textstyle\overset{(D)}{=}\frac{1}{9}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt[4]{16x^8}\cloze{\overset{(H)}{=}2x^2}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{2}{3})^3\cloze{\textstyle\overset{(G)}{=}\frac{8}{27}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^3\cdot a^6\cloze{\overset{(A)}{=}a^9}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (x^2)^4\cloze{\overset{(C)}{=}x^8}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3^x\cdot 2^x\cloze{\overset{(B)}{=}6^x}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 8^\frac{1}{3}\cloze{\small\overset{(E)}{=}\sqrt[3]{8}=2}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 5^4:5^2\cloze{\overset{(I)}{=}5^2=25}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt[3]{64x^8}\cloze{\small\overset{(H)}{=}4\sqrt[3]{x^8}\overset{(E)/(C)}{=}4x^\frac{8}{3}}$

Aus diesen Regeln kann man weitere Regeln herleiten:

F) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^0=\cloze{1}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{8pt}a^1=\cloze{a}$
G) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{a}{b})^n=\cloze{\textstyle\frac{a^n}{b^n}}$
H) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt[n]{a\cdot b}=\cloze{\sqrt[n]{a}\cdot\sqrt[n]{b}}$
I) $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} a^n:a^m=\cloze{a^{n-m}}$

Berechnen Sie ohne Hilfsmittel!

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^2$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-3)^4$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^5$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 0{,}8^2$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^3$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 0{,}9^2$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-2)^5$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-2)^3$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 1{,}3^2$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-3)^3$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^2$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (-2)^2$

Quadratzahlen

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 1^2=1\hspace{22pt} 11^2=121$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 2^2=4\hspace{22pt} 12^2=144$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 3^2=9\hspace{22pt} 13^2=169$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 4^2=16\hspace{17pt} 14^2=196$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 5^2=25\hspace{17pt} 15^2=225$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 6^2=36\hspace{17pt} 16^2=256$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 7^2=49\hspace{17pt} 17^2=289$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 8^2=64\hspace{17pt} 18^2=324$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt} 9^2=81\hspace{17pt} 19^2=361$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 10^2=100\hspace{12pt} 20^2=400$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 5^{-3}\cloze{\textstyle=\frac{1}{125}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^{-2}\cloze{\textstyle=\frac{1}{16}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{4})^{-2}\cloze{\textstyle=16}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{5})^{-2}\cloze{\textstyle=25}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^{-3}\cloze{\textstyle=\frac{1}{64}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{2})^{-2}\cloze{\textstyle=4}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{2})^{-3}\cloze{\textstyle=8}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{4})^{-4}\cloze{\textstyle=256}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3^{-1}\cloze{\textstyle=\frac{1}{3}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 4^{-1}\cloze{\textstyle=\frac{1}{4}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{4})^{-3}\cloze{\textstyle=64}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{1}{5})^{-1}\cloze{\textstyle=5}$

Kubikzahlen

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}1^3=1\hspace{22pt} 11^3=1331$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}2^3=8\hspace{22pt} 12^3=1728$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}3^3=27\hspace{17pt} 13^3=2197$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}4^3=64\hspace{17pt} 14^3=2744$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}5^3=125\hspace{12pt} 15^3=3375$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}6^3=216\hspace{12pt} 16^3=4096$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}7^3=343\hspace{12pt} 17^3=4913$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}8^3=512\hspace{12pt} 18^3=5832$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{5pt}9^3=729\hspace{12pt} 19^3=6859$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 10^3=1000\hspace{7pt} 20^3=8000$

Erklären Sie, welche häufigen Fehler hier jeweils gemacht wurden und korrigieren Sie.

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^{-3}=-8$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \textrm{}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt{4x^6}=2x^6$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^3\cdot 2^4=4^7$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \textrm{}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (4^3)^2=4^9$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^3=6$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \textrm{}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 2^3\cdot 2^4=2^{12}$

Berechnen Sie im Kopf!

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3^2\cdot 6^2\cloze{=18^2=324}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 3^5:3^3\cloze{=3^{5-3}=3^2=9}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 12^{-2}\cloze{\textstyle=\frac{1}{12^2}=\frac{1}{144}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 0{,}04^3\cloze{\textstyle=\left(\frac{4}{100}\right)^3}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \cloze{\textstyle=\frac{64}{1000000}=0{,}000064}$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} 0{,}25^{-0{,}5}\cloze{\small\textstyle=\sqrt{\frac{1}{4}}^{-1}=\left(\frac{1}{2}\right)^{-1}=2}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt[3]{13^6}\cloze{\textstyle=13^{\frac{6}{3}}=13^2=169}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} (\frac{3}{4})^2\cloze{\textstyle=\frac{3^2}{4^2}= \frac{9}{16}}$

Auch wenn man einen Ausdruck nicht komplett ohne Wurzel schreiben kann, kann es helfen, die Wurzel teilweise zu ziehen.
Bsp:

\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt{72}-\sqrt{2}=\sqrt{9\cdot4\cdot2}-\sqrt{2}=3\cdot2\cdot\sqrt{2}-\sqrt{2}=6\sqrt{2}-\sqrt{2}=5\sqrt{2}

Vereinfachen Sie entsprechend!

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt[3]{54}\cloze{\textstyle=\sqrt[3]{2\cdot27}=3\sqrt[3]{2}}$
$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt{\frac{1}{2}}+\sqrt{2}\cloze{\small\textstyle=\sqrt{\frac{2}{4}}+\sqrt{2}}$ $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \cloze{\textstyle=\frac{\sqrt{2}}{2}+\sqrt{2}=\frac{3}{2}\sqrt{2}}$

$\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \sqrt[3]{4}\cdot\sqrt{2}\cloze{\textstyle=2^\frac{2}{3}\cdot2^\frac{1}{2}=2^\frac{5}{6}}$

Zum Weiterlesen: $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{3cm}$

Primfaktorzerlegung $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{2.6cm}$

https://youtu.be/6gXQUCzv9hM $\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}}}} \hspace{0.95cm}$