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[latex]\sqrt[]{a\cdot b}=\sqrt[]{a}\cdot\sqrt[]{b},\hspace{2mm}a\ge0,b\ge0[/latex]

[latex]\sqrt[3]{a\cdot b}=\sqrt[3]{a}\cdot\sqrt[3]{b}[/latex]

[latex]\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}[/latex]

Korijen kvocijenta

[latex]\sqrt[]{\frac{a}{b}}=\frac{\sqrt[]{a}}{\sqrt[]{b}},\hspace{2mm}a\ge0, b>0[/latex] 

[latex]\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}}, \hspace{2mm}b\neq0[/latex]

[latex]\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}, \hspace{2mm}b\neq0[/latex]

Vrijedi:

[latex]\sqrt[]{a^2}=\lvert{a}\rvert,\hspace{2mm}\sqrt[3]{a^3}=a[/latex]

Primjer 1

Pojednostavimo.

a)  [latex]\sqrt[3]{-64a^3}[/latex]

b)  [latex]\sqrt[]{\frac{x^2}{4}}[/latex]

c)  [latex]\sqrt[]{x^2-2x+1}[/latex]

d)  [latex]\sqrt[3]{-8(x+3)^3}[/latex]

Rješenje

a) [latex]\sqrt[3]{-64a^3}=[/latex]

[latex]=\sqrt[3]{-64}\cdot\sqrt[3]{a^3}[/latex]

[latex]=-4a[/latex]

b) [latex]\sqrt[]{\frac{x^2}{4}}=\frac{\sqrt[]{x^2}}{\sqrt[]{4}}=\frac{\lvert{x}\rvert}{2}[/latex]

Ako je [latex]x>0[/latex], onda je [latex]\sqrt[]{\frac{x^2}{4}}=\frac{\sqrt[]{x^2}}{\sqrt[]{4}}=\frac{x}{2}[/latex] jer je tada [latex]\lvert{x}\rvert=x[/latex].

c) [latex]\sqrt[]{x^2-2x+1}=[/latex]

[latex]=\sqrt[]{(x-1)^2}[/latex]

[latex]=\lvert{x-1}\rvert[/latex]

 

d) [latex]\sqrt[3]{-8(x+3)^3}=[/latex]

[latex]=\sqrt[3]{-8}\cdot\sqrt[3]{(x+3)^3}[/latex]

[latex]=-2(x+3)[/latex]

U sljedećim primjerima i zadatcima uzimamo da sve varijable imaju pozitivnu vrijednost.

To znači da umjesto npr. [latex]\lvert{x}\rvert[/latex] možemo pisati samo [latex]x[/latex].

Zadatak 1

Primjer 2

Pojednostavimo.

a)   [latex]\sqrt[]{20}+\sqrt[]{5}[/latex]

b)   [latex]\sqrt[3]{24x}-\sqrt[3]{81x}[/latex]

c)   [latex]\sqrt[]{30x^3y}\cdot\sqrt[]{\frac{3}{5}xy}[/latex]

d)   [latex]\sqrt[3]{128x^8y^4}\colon\sqrt[3]{2x^2y^2}[/latex]

 

Rješenje

a) [latex]\sqrt{20}+\sqrt{5}=[/latex]

[latex]=\sqrt{4\cdot5}+\sqrt{5}[/latex]

[latex]=\sqrt{4}\cdot\sqrt{5}+\sqrt{5}[/latex]

[latex]=2\sqrt{5}+\sqrt{5}[/latex]

[latex]=3\sqrt{5}[/latex]

 

b) [latex]\sqrt[3]{24x}-\sqrt[3]{81x}=[/latex]

[latex]=\sqrt[3]{8\cdot3x}-\sqrt[3]{27\cdot3x}[/latex]

[latex]=2\sqrt[3]{3x}-3\sqrt[3]{3x}[/latex]

[latex]=-\sqrt[3]{3x}[/latex]

c) [latex]\sqrt[]{30x^3y}\cdot\sqrt[]{\frac{3}{5}xy}=[/latex]

[latex]=\sqrt[]{30x^3y\cdot\frac{3}{5}xy}[/latex]

[latex]=\sqrt[]{18x^4y^2}[/latex]

[latex]=3x^2y\sqrt{2}[/latex]

d) [latex]\sqrt[3]{128x^8y^4}\colon\sqrt[3]{2x^2y^2}=[/latex]

[latex]=\sqrt[3]{128x^8y^4\colon2x^2y^2}[/latex]

[latex]=\sqrt[3]{64x^6y^2}[/latex]

[latex]=4x^2\sqrt[3]{y^2}[/latex]

Zadatak 2

Primjer 3

Racionalizirajmo nazivnike razlomaka.

a)   [latex]\frac{2}{\sqrt[]{5}}[/latex]

b)   [latex]\frac{15}{\sqrt[3]{5a^2}}[/latex]

c)   [latex]\frac{2}{\sqrt[]{7}-1}[/latex]

d)   [latex]\frac{a-b}{\sqrt{a}-\sqrt{b}}[/latex]

e)   [latex]\frac{1}{\sqrt[3]{2}+\sqrt[3]{3}}[/latex]

f)   [latex]\frac{2}{\sqrt[3]{25}+\sqrt[3]{5}+1}[/latex]

 

Rješenje

a) [latex]\frac{2}{\sqrt{5}}=\frac{2}{\sqrt{5}}\cdot\frac{\sqrt{5}}{\sqrt{5}}[/latex]

    [latex]=\frac{2\sqrt{5}}{\left(\sqrt{5}\right)^2}=\frac{2\sqrt{5}}{5}[/latex]

 

b) [latex]\frac{15}{\sqrt[3]{5a^2}}=\frac{15}{\sqrt[3]{5a^2}}\cdot\frac{\sqrt[3]{25a}}{\sqrt[3]{25a}}[/latex]

       [latex]=\frac{15\sqrt[3]{25a}}{\sqrt[3]{125a^3}}[/latex]

       [latex]=\frac{15\sqrt[3]{25a}}{5a}[/latex]

       [latex]=\frac{3\sqrt[3]{25a}}{a}[/latex]

c) [latex]\frac{2}{\sqrt{7}-1}=[/latex]

           [latex]=\frac{2}{\sqrt{7}-1}\cdot\frac{\sqrt{7}+1}{\sqrt{7}+1}[/latex]

            [latex]=\frac{2\cdot\left(\sqrt{7}+1\right)}{\left(\sqrt{7}\right)^2-1}[/latex]

     [latex]=\frac{2\cdot(\sqrt[]{7}+1)}{7-1}[/latex]

     [latex]=\frac{2\cdot(\sqrt[]{7}+1)}{6}[/latex]

     [latex]=\frac{(\sqrt[]{7}+1)}{3}[/latex]

d) [latex]\frac{a-b}{\sqrt{a}-\sqrt{b}}=[/latex]

  [latex]=\frac{a-b}{\sqrt{a}-\sqrt{b}}\cdot\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}[/latex]

  [latex]=\frac{(a-b)\left(\sqrt{a}+\sqrt{b}\right)}{\left(\sqrt{a}\right)^2-\left(\sqrt{b}\right)^2}[/latex]

  [latex]=\frac{(a-b)\left(\sqrt{a}+\sqrt{b}\right)}{a-b}[/latex]

  [latex]=\sqrt{a}+\sqrt{b}[/latex]

 

e) [latex]\frac{1}{\sqrt[3]{2}+\sqrt[3]{3}}=[/latex]

        [latex]=\frac{1}{\sqrt[3]{2}+\sqrt[3]{3}}\cdot\frac{\left(\sqrt[3]{2}\right)^2-\sqrt[3]{2}\cdot\sqrt[3]{3}+\left(\sqrt[3]{3}\right)^2}{\left(\sqrt[3]{2}\right)^2-\sqrt[3]{2}\cdot\sqrt[3]{3}+\left(\sqrt[3]{3}\right)^2}[/latex]

        [latex]=\frac{\left(\sqrt[3]{2}\right)^2-\sqrt[3]{2}\cdot\sqrt[3]{3}+\left(\sqrt[3]{3}\right)^2}{\left(\sqrt[3]{2}\right)^3+\left(\sqrt[3]{3}\right)^3}[/latex]

        [latex]=\frac{\left(\sqrt[3]{2}\right)^2-\sqrt[3]{2}\cdot\sqrt[3]{3}+\left(\sqrt[3]{3}\right)^2}{5}[/latex]

 

f) [latex]\frac{2}{\sqrt[3]{25}+\sqrt[3]{5}+1}=[/latex]

     [latex]=\frac{2}{\sqrt[3]{25}+\sqrt[3]{5}+1}\cdot\frac{\sqrt[3]{5}-1}{\sqrt[3]{5}-1}[/latex]

     [latex]=\frac{2\cdot\left(\sqrt[3]{5}-1\right)}{\left(\sqrt[3]{5}\right)^3-1}[/latex]

     [latex]=\frac{2\cdot\left(\sqrt[3]{5}-1\right)}{4}[/latex]

     [latex]=\frac{\sqrt[3]{5}-1}{2}[/latex]

 

Zadatak 3

Primjer 4

Pojednostavimo zadane izraze.

a)    [latex]\sqrt[4]{9x^6}\cdot\sqrt[4]{9x^2}[/latex] 

b)    [latex]\frac{\sqrt[5]{3a^{13}}}{\sqrt[5]{96a^3}}[/latex]

c)    [latex]2x\sqrt[5]{96x}+5\sqrt[5]{3x^6}[/latex]

Rješenje

a) [latex]\sqrt[4]{9x^6}\cdot\sqrt[4]{9x^2}=[/latex]

[latex]=\sqrt[4]{9x^6\cdot9x^2}[/latex]

[latex]=\sqrt[4]{81x^8}[/latex]

[latex]=\sqrt[4]{\left(3x^2\right)^4}[/latex]

[latex]=3x^2[/latex]

 

b) [latex]\frac{\sqrt[5]{3a^{13}}}{\sqrt[5]{96a^3}}=[/latex]

   [latex]=\sqrt[5]{\frac{3a^{13}}{96a^3}}[/latex]

   [latex]=\sqrt[5]{\frac{a^{10}}{32}}[/latex]

   [latex]=\sqrt[5]{\left(\frac{a^2}{2}\right)^5}[/latex]

   [latex]=\frac{a^2}{2}[/latex]

 

c) [latex]2x\sqrt[5]{96x}+5\sqrt[5]{3x^6}=[/latex]

    [latex]=2x\sqrt[5]{32\cdot3\cdot x}+5\sqrt[5]{3\cdot x^5\cdot x}[/latex]

    [latex]=4x\sqrt[5]{3x}+5x\sqrt[5]{3x}[/latex]

    [latex]=9x\sqrt[5]{3x}[/latex]

Zadatak 4

Pojednostavite zadane izraze.  

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