Działania na pierwiastkach

Działania na pierwiastkach

• Działania na pierwiastkach:
  • dodawanie pierwiastków
    \textcolor{#75A600}{4}\sqrt[{\textcolor{#005FD4}{4}}]{\textcolor{#E94D00}{5}}+\textcolor{#75A600}{5}\sqrt[{\textcolor{#005FD4}{4}}]{\textcolor{#E94D00}{5}}=\left(\textcolor{#75A600}{4}+\textcolor{#75A600}{5}\right)\sqrt[{\textcolor{#005FD4}{4}}]{\textcolor{#E94D00}{5}}= 44√5 + 54√5  =  (4 + 5)4√5  =
    =\textcolor{#75A600}{9}\sqrt[{\textcolor{#005FD4}{4}}]{\textcolor{#E94D00}{5}} =  94√5
  • odejmowanie pierwiastków
    \textcolor{#75A600}{3}\sqrt[{\textcolor{#005FD4}{5}}]{\textcolor{#E94D00}{6}}-\sqrt[{\textcolor{#005FD4}{5}}]{\textcolor{#E94D00}{6}}=\left(\textcolor{#75A600}{3}-\textcolor{#75A600}{1}\right)\sqrt[{\textcolor{#005FD4}{5}}]{\textcolor{#E94D00}{6}}= 35√6 − 5√6  =  (3 − 1)5√6  =
    =\textcolor{#75A600}{2}\sqrt[{\textcolor{#005FD4}{5}}]{\textcolor{#E94D00}{6}} =  25√6
  • mnożenie pierwiastków
    \sqrt[{3}]{-2}\cdot \sqrt[{3}]{4}=\sqrt[{3}]{-2\cdot 4}= 3√−2 ⋅ 3√4  =  3√−2 ⋅ 4  =
    =\sqrt[{3}]{-8}=-2 =  3√−8  =  −2
    \sqrt[{\textcolor{#E94D00}{n}}]{a}\cdot \sqrt[\textcolor{#E94D00}{{n}}]{b}=\sqrt[\textcolor{#E94D00}{{n}}]{a\cdot b} n√a ⋅ n√b  =  n√a ⋅ b
  • dzielenie pierwiastków
    \sqrt[{6}]{128}:\sqrt[{6}]{2}=\sqrt[{6}]{128:2}= 6√128  :  6√2  =  6√128  :  2  =
    =\sqrt[{6}]{64}=2 =  6√64  =  2
    \frac{\sqrt[{\textcolor{#E94D00}{n}}]{a}}{\sqrt[\textcolor{#E94D00}{{n}}]{b}}=\sqrt[\textcolor{#E94D00}{{n}}]{\frac{a}{b}} n√an√b  =  n√ab
  • wyłączanie czynnika przed pierwiastek
    \sqrt{18}=\sqrt{\left.\mleft.3\mright.\right.^{\mleft.2\mright.}\cdot 2}=3\sqrt{2} √18  =  √32 ⋅ 2  =  3√2
  • włączanie czynnika pod pierwiastek
    2\sqrt[{3}]{4}=\sqrt[{3}]{\left.\mleft.2\mright.\right.^{\mleft.3\mright.}\cdot 4}=\sqrt[{3}]{32} 23√4  =  3√23 ⋅ 4  =  3√32
• Usuwanie niewymierności z mianownika
  \frac{5}{\sqrt{3}}=\frac{5}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{5\sqrt{3}}{3} 5√3  =  5√3 ⋅ √3√3  =  5√33
  
  \frac{k}{\sqrt{a}}=\frac{k}{\sqrt{a}}\cdot \frac{\sqrt{a}}{\sqrt{a}}=\frac{k\sqrt{a}}{a} k√a  =  k√a ⋅ √a√a  =  k√aa
  
  
• Pierwiastek z sumy to nie jest to samo, co suma pierwiastków
  \sqrt{\left.\mleft.6\mright.\right.^{\mleft.2\mright.}+\left.\mleft.8\mright.\right.^{\mleft.2\mright.}}\neq \sqrt{\left.\mleft.6\mright.\right.^{\mleft.2\mright.}}+\sqrt{\left.\mleft.8\mright.\right.^{\mleft.2\mright.}} √62 + 82  ≠  √62 + √82
  
  \sqrt{\left.\mleft.25\mright.\right.^{\mleft.2\mright.}-\left.\mleft.15\mright.\right.^{\mleft.2\mright.}}\neq \sqrt{\left.\mleft.25\mright.\right.^{\mleft.2\mright.}}-\sqrt{\left.\mleft.15\mright.\right.^{\mleft.2\mright.}} √252 − 152  ≠  √252 − √152
  
  \sqrt[{\textcolor{#E94D00}{n}}]{a+b}\neq \sqrt[{\textcolor{#E94D00}{n}}]{a}+\sqrt[{\textcolor{#E94D00}{n}}]{b} n√a + b  ≠  n√a + n√b