Esercizi sulla potenza e radice di radice

Esercizio n° 1

1) $(\sqrt[4]{2}) ^{2}$ ; $(\sqrt[3]{3}) ^{6}$ ; $(\sqrt[9]{3}) ^{3}$

2) $(\frac{2}{3}\sqrt{3}) ^{3}$ ; $(2\sqrt{\frac{1}{2}}) ^{2}$ ; $(\sqrt[6]{2xy^{2}})^{3}$ con x≥0

3) $(a\sqrt{a+2b}) ^{3}$ ; $(\frac{a+2}{a+1}\sqrt{\frac{a+1}{a+2}}) ^{2}$

Esercizio n° 2

Semplifica le seguenti espressioni con potenze di radicali.

4) $(\sqrt[6]{1-\frac{x-3y}{x+y}}\cdot \sqrt[6]{\frac{x-y}{4y}}) ^{3}$ : $\sqrt{\frac{1}{x+y}}$

5) $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)}$ $\cdot \sqrt[3]{\frac{x}{y}-2+\frac{y}{x}}$ : $(\sqrt[3]{x ^{3}-2x ^{2}y+xy ^{2}}) ^{2}$

6) $(\sqrt[3]{ a^{2}-4a+4 }) ^{2}: \sqrt[3]{ a^{2}+2a }: [ (\sqrt[6]{\frac{a-2}{a+2}}) ^{5} : \sqrt[]{\frac{a}{a^{2}-4}}]$

Esercizio n° 4

Esegui le seguenti radici di radici.

7) $\sqrt[6]{\sqrt{3}}$ ; $\sqrt{\sqrt[3]{3a ^{2}b ^{3}}}$ ; $\sqrt[3]{\sqrt{\sqrt{\frac{1}{2}}}}$

8) $\sqrt{a\sqrt[4]{a}}$ ; $\sqrt[3]{x\sqrt{x\sqrt[3]{x\sqrt{x}}}}$ ; $\sqrt[3]{2\sqrt{\frac{1}{2}\sqrt[3]{\frac{1}{4}}}}$

9) $\sqrt[5]{ a ^{4}\sqrt[3]{\frac{1}{a ^{2}}}}}$ ; $\sqrt[]{x\sqrt[3]{x}}\cdot\sqrt[3]{x\sqrt[]{x}} \cdot\sqrt[3]{x\sqrt[3]{x}}$ con x≥o

SVOLGIMENTO

Esercizio n° 1

Esegui le seguenti potenze di radicali.

1) $(\sqrt[4]{2}) ^{2}$ ; $(\sqrt[3]{3}) ^{6}$ ; $(\sqrt[9]{3}) ^{3}$

Applichiamo il teorema della potenza: $(\sqrt[n]{a}) ^{m} = \sqrt[n]{a ^{m} }$

$(\sqrt[4]{2}) ^{2}$ = $\sqrt[4]{2^{2}}$ = $\sqrt[]{2}$
$(\sqrt[3]{3}) ^{6}$ = $\sqrt[3]{3^{6}}$ = semplificando l’indice con l’esponente del radicando otteniamo: 9
$(\sqrt[9]{3}) ^{3}$ = $\sqrt[9]{3^{3}$ = $\sqrt[3]{3$

2) $(\frac{2}{3}\sqrt{3}) ^{3}$ ; $(2\sqrt{\frac{1}{2}}) ^{2}$ ; $(\sqrt[6]{2xy^{2}})^{3}$ con x≥0

$(\frac{2}{3}\sqrt{3}) ^{3}$ = $\frac{8}{27}\sqrt{3^{3}} = \frac{8}{27} \cdot \sqrt{3^{2} \cdot 3} = \frac{8}{27} \cdot3 \sqrt{3} = \frac{8}{9}\sqrt{3}$

$(2\sqrt{\frac{1}{2}}) ^{2}$ = $4\sqrt{\frac{1}{2} ^{2}}$ = $4\sqrt{\frac{1}{4}}$ = $4\sqrt{\frac{1}{2 ^{2}}}$ = $\frac{4}{2}$ = 2

$(\sqrt[6]{2xy^{2}})^{3}$ = $\sqrt[6]{2^{3} x^{3} y^{6}}$ = $\sqrt[2]{2 xy^{2}}$ = $\left \| y|\sqrt[2]{2 x}$

3) $(a\sqrt{a+2b}) ^{3}$ ; $(\frac{a+2}{a+1}\sqrt{\frac{a+1}{a+2}}) ^{2}$

$(a\sqrt{a+2b}) ^{3}$ = $a^{3}\sqrt{(a+2b)^{3}}$ = $a^{3}\sqrt{(a+2b)^{2}(a+2b)}$ =

= $a^{3}(a+2b)\sqrt{(a+2b)}$

$(\frac{a+2}{a+1}\sqrt{\frac{a+1}{a+2}}) ^{2}$ = $\frac{(a+2) ^{2}}{(a+1) ^{2}}\sqrt{\frac{(a+1) ^{2}}{(a+2) ^{2}}}$ =

= $\frac{(a+2) ^{2}}{(a+1) ^{2}} \cdot \frac{(a+1)}{(a+2)}$ semplificando si ottiene $\frac{(a+2) }{(a+1) }$

Esercizio n° 2

Semplifica le seguenti espressioni con potenze di radicali.

4) $(\sqrt[6]{1-\frac{x-3y}{x+y}}\cdot \sqrt[6]{\frac{x-y}{4y}}) ^{3}$ : $\sqrt{\frac{1}{x+y}}$ = $\sqrt[6]{(1-\frac{x-3y}{x+y})^{3}}\cdot \sqrt[6]{(\frac{x-y}{4y}) ^{3} }: \sqrt{\frac{1}{x+y}}$ =

= $\sqrt[6]{(\frac{x+y -(x-3y)}{x+y})^{3}}\cdot \sqrt[6]{(\frac{x-y}{4y}) ^{3} }: \sqrt{\frac{1}{x+y}} = \sqrt[6]{(\frac{x+y -x+3y}{x+y})^{3}}\cdot \sqrt[6]{(\frac{x-y}{4y}) ^{3} }: \sqrt{\frac{1}{x+y}}$ = semplifichiamo il 6 dell’indice con l’esponente del radicando.

= $\sqrt[]{(\frac{4y}{x+y})}\cdot \sqrt[]{(\frac{x-y}{4y}) }: \sqrt{\frac{1}{x+y}} = \sqrt[]{(\frac{4y}{x+y})\cdot(\frac{x-y}{4y}): \frac{1}{x+y} } =$ $\sqrt[]{(\frac{4y}{x+y})\cdot(\frac{x-y}{4y})\cdot (x+y)$

= semplifico (x+y) e poi 4y ottengo: $\sqrt[]{x-y}$ .

5) $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)}$ $\cdot \sqrt[3]{\frac{x}{y}-2+\frac{y}{x}}$ : $(\sqrt[3]{x ^{3}-2x ^{2}y+xy ^{2}}) ^{2}$

= $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)} \cdot \sqrt[3]{\frac{x}{y}-2+\frac{y}{x}} : \sqrt[3]{[x(x ^{2}-2x y+y ^{2})} ]^{2}$

= $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)} \cdot \sqrt[3]{\frac{x}{y}-2+\frac{y}{x}} : \sqrt[3]{x^{2}(x -y) ^{4}} =$

= $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)} \cdot \sqrt[3]{\frac{x ^{2}-2xy+y ^{2}}{xy}} ^{} : \sqrt[3]{x^{2}(x -y) ^{4}} =$

= $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)} \cdot \sqrt[3]{\frac{(x -y) ^{2}}{xy}} : \sqrt[3]{x^{2}(x -y) ^{4}}=$

= $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)\cdot\frac{(x -y) ^{2}}{xy}:x^{2}(x -y) ^{4} } =$

= $\sqrt[3]{x^{2}y(x-y) ^{2}(x+y)\cdot\frac{(x -y) ^{2}}{xy} \cdot \frac{1}{x^{2}(x -y) ^{4} } }$ = semplificando tutto rimane $\sqrt[3]{ \frac{(x+y)}{x} }$

6) $(\sqrt[3]{ a^{2}-4a+4 }) ^{2}: \sqrt[3]{ a^{2}+2a }: [ (\sqrt[6]{\frac{a-2}{a+2}}) ^{5} : \sqrt[]{\frac{a}{a^{2}-4}}]$ =

= $\sqrt[3]{( a^{2}-4a+4 ) ^{2}}: \sqrt[3]{ a^{2}+2a }: [ \sqrt[6]{\frac{(a-2)^{5}}{(a+2)^{5}}} : \sqrt[]{\frac{a}{a^{2}-4}}]$ = porto l’ultima radice ad indice 6 in kodo che si possano svolgere i calcoli

= $\sqrt[3]{( a-2 ) ^{4}}: \sqrt[3]{ a^{2}+2a }: [ \sqrt[6]{\frac{(a-2)^{5}}{(a+2)^{5}}} : \sqrt[6]{\frac{a^{3}}{(a^{2}-4)^{3}}}]$ =

= $\sqrt[3]{( a-2 ) ^{4}}: \sqrt[3]{ a^{2}+2a }: [ \sqrt[6]{\frac{(a-2)^{5}}{(a+2)^{5}}\cdot\frac{(a^{2}-4)^{3}}{a^{3}}} ]$ =

= $\sqrt[3]{( a-2 ) ^{4}}: \sqrt[3]{ a(a+2) }: \sqrt[6]{\frac{(a-2)^{5}}{(a+2)^{5}}\cdot\frac{(a-2)^{3}(a+2)^{3}}{a^{3}}}$ = portiamo tutti allo stesso indice

= $\sqrt[6]{( a-2 ) ^{8}}: \sqrt[6]{ a^{2}(a+2)^{2} }: \sqrt[6]{\frac{(a-2)^{5}}{(a+2)^{5}}\cdot\frac{(a-2)^{3}(a+2)^{3}}{a^{3}}}$ =

= $\sqrt[6]{( a-2 ) ^{8} \cdot \frac{1}{a^{2}(a+2)^{2}}\cdot \frac{(a+2)^{5}}{(a-2)^{5}}\cdot\frac{ a^{3} }{(a-2)^{3}(a+2)^{3}}}$ = semplificando tutto rimane

= $\sqrt[6]{a}$

Esercizio n° 4

Esegui le seguenti radici di radici.

7) $\sqrt[6]{\sqrt{3}}$ ; $\sqrt{\sqrt[3]{3a ^{2}b ^{3}}}$ ; $\sqrt[3]{\sqrt{\sqrt{\frac{1}{2}}}}$

$\sqrt[6]{\sqrt{3}}$ = si moltiplicano gli indice delle radici ottenendone una sola

= $\sqrt[12]{3}$

$\sqrt{\sqrt[3]{3a ^{2}b ^{3}}}$ = $\sqrt[6]{3a ^{2}b ^{3}}}$
$\sqrt[3]{\sqrt{\sqrt{\frac{1}{2}}}}$ = $\sqrt[12]{\frac{1}{2}}$

8) $\sqrt{a\sqrt[4]{a}}$ ; $\sqrt[3]{x\sqrt{x\sqrt[3]{x\sqrt{x}}}}$ ; $\sqrt[3]{2\sqrt{\frac{1}{2}\sqrt[3]{\frac{1}{4}}}}$

$\sqrt{a\sqrt[4]{a}}$ = $\sqrt{\sqrt[4]{a^{4} \cdot a} }$ = $\sqrt[8]{a^{4} \cdot a}$ = $\sqrt[8]{a^{5} }$
$\sqrt[3]{x\sqrt{x\sqrt[3]{x\sqrt{x}}}}$ = $\sqrt[3]{\sqrt{x^{2} \cdot x\sqrt[3]{x\sqrt{x}}}}$ = $\sqrt[3]{\sqrt{x^{3} \sqrt[3]{x\sqrt{x}}}}$ = $\sqrt[3]{\sqrt{\sqrt[3]{x^{9} \cdot x\sqrt{x}}}}$ =

= $\sqrt[3]{\sqrt{\sqrt[3]{x^{10} \sqrt{x}}}}$ = $\sqrt[3]{\sqrt{\sqrt[3]{x^{10} \sqrt{x^{20} \cdot x}}}}$ = $\sqrt[3]{\sqrt{\sqrt[3]{ \sqrt{x^{21} }}}}$ = $\sqrt[36]{x^{21}}$ divido indice della radice ed esponente del radicando per tre.

= $\sqrt[12]{x^{7}}$

$\sqrt[3]{2\sqrt{\frac{1}{2}\sqrt[3]{\frac{1}{4}}}}$ = $\sqrt[3]{\sqrt{4\cdot\frac{1}{2}\sqrt[3]{\frac{1}{4}}}}$ = $\sqrt[3]{\sqrt{\sqrt[3]{8 \cdot\frac{1}{4}}}}$ =

= $\sqrt[3]{\sqrt{\sqrt[3]{2}}}$ = $\sqrt[18]{2}$

9) $\sqrt[5]{ a ^{4}\sqrt[3]{\frac{1}{a ^{2}}}}}$ ; $\sqrt[]{x\sqrt[3]{x}}\cdot\sqrt[3]{x\sqrt[]{x}} \cdot\sqrt[3]{x\sqrt[3]{x}}$ con x≥o

$\sqrt[5]{ a ^{4}\sqrt[3]{\frac{1}{a ^{2}}}}}$ = $\sqrt[5]{\sqrt[3]{ a ^{4} \cdot \frac{1}{a ^{2}}}}}$ = $\sqrt[5]{\sqrt[3]{ a ^{2} }}}$ =

= $\sqrt[15]{a ^{2}}}$

$\sqrt[]{x\sqrt[3]{x}}\cdot\sqrt[3]{x\sqrt[]{x}} \cdot\sqrt[3]{x\sqrt[3]{x}}$ con x≥o

= $\sqrt[]{\sqrt[3]{x ^{3} \cdot x}}\cdot\sqrt[3]{\sqrt[]{x ^{2} \cdot x}} \cdot\sqrt[3]{\sqrt[3]{x ^{3} \cdot x}}$ = $\sqrt[]{\sqrt[3]{x ^{4} }}\cdot\sqrt[3]{\sqrt[]{x ^{3} }} \cdot\sqrt[3]{\sqrt[3]{x ^{4}}}$ =