Esercizi proprietà invariantiva radicali

Esercizio n° 1

Semplifica i seguenti radicali.

1) $\sqrt[10]{32}$ ; $\sqrt[4]{9}$ ; $\sqrt[6]{25}$

2) $\sqrt[6]{27a ^{3}b ^{6}}$ con a>0; $\sqrt[10]{32a ^{5}b ^{5}}$ a·b>0

3) $\sqrt[20]{x ^{10}y ^{5}z ^{5}t ^{20}}$ y ·z>0

4) $\sqrt[]{a ^{4}b ^{6}}$ ; $\sqrt[]{a ^{2}b ^{4}}$ ; $\sqrt[3]{a ^{6}b ^{9}}$ b>0

5) $\sqrt[6]{a ^{4}-8a^{2}b ^{2}+ 16b^{4}}$

6) $\sqrt[4]{a ^{2}+2ab +b ^{2}}$

7) $\sqrt[6]{\frac{8a ^{3}+ 24a ^{2}b+24a b ^{2}+ 8b ^{3}}{(27a ^{3}-81a^{2}b +81a b ^{2}-27b ^{3})(x ^{3}+ 3x ^{2}y +3xy ^{2}+y^{3})}}$

8) $\sqrt[4]{\frac{c ^{4}a ^{2}(x -2z)^{2}}{4(a -b)^{2}}}$

9) $\sqrt[3n]{\frac{x ^{n}a ^{9n}}{b^{2n}}}$

Esercizio n° 2

Ridurre allo stesso indice i seguenti radicali

10) $\sqrt[3]{5}$ ; $\sqrt{2}$ ; $\sqrt[4]{7}$ ; $\sqrt[6]{3}$

11) $\sqrt[6]{(a-b) ^{2}}$ ; $\sqrt[]{(a+b) }$ ; $\sqrt[3]{(a+b) }$

12) $\sqrt[7]{ 2x ^{6}}$ ; $\sqrt[]{ x y}$ ; $\sqrt[14]{ x y ^{7}}$

13) $\sqrt[]{\frac{2xy}{x+1}}$ ; $\sqrt[6]{\frac{xz}{2x - 1}}$ ; $\sqrt[4]{\frac{x^{3}y}{3}} ;$

SVOLGIMENTO

Esercizio n° 1

Semplifica i seguenti radicali.

1) $\sqrt[10]{32}$ ; $\sqrt[4]{9}$ ; $\sqrt[6]{25}$

$\sqrt[10]{32}$ = $\sqrt[10]{2 ^{5}}$ = $\sqrt[10: 5]{2 ^{5: 5}}$ = $\sqrt{2}$

$\sqrt[4]{9}$ = $\sqrt[4]{3 ^{2}}$ = $\sqrt[4: 2]{3 ^{2: 2}}$ = $\sqrt{3}$

2) $\sqrt[6]{27a ^{3}b ^{6}}$ con a>0; $\sqrt[10]{32a ^{5}b ^{5}}$ a·b>0

$\sqrt[6]{27a ^{3}b ^{6}}$ con a>0 = $\sqrt[6]{3 ^{3}a ^{3}b ^{6}}$ = $\sqrt[6]{(3 ab^{2} )^{3}}$ = $\sqrt[]{3 ab^{2} }$

$\sqrt[10]{32a ^{5}b ^{5}}$ a·b>0 = $\sqrt[10]{2^{5}a ^{5}b ^{5}}$ = $\sqrt[10]{(2a b) ^{5}}$ = $\sqrt[]{2a b}$

3) $\sqrt[20]{x ^{10}y ^{5}z ^{5}t ^{20}}$ y ·z>0

$\sqrt[20]{x ^{10}y ^{5}z ^{5}t ^{20}}$ = $\sqrt[20]{(x ^{2}y z t ^{4}) ^{5}}$ = $\sqrt[4]{x ^{2}y z t ^{4} }$

4) $\sqrt[]{a ^{4}b ^{6}}$ ; $\sqrt[]{a ^{2}b ^{4}}$ ; $\sqrt[3]{a ^{6}b ^{9}}$ b>0

$\sqrt[]{a ^{4}b ^{6}}$ = $\sqrt[]{(a ^{2}b ^{3})^{2}}$ = $a ^{2}\left \| b| ^{3}$ si mette il valore assoluto perchè la b deve essere per forza positiva

$\sqrt[]{a ^{2}b ^{4}}$ = $\sqrt[]{(a b ^{2})^{2}}$ = $\left \| a| b^{2}$

$\sqrt[3]{a ^{6}b ^{9}}$ = $\sqrt[3]{(a ^{2}b ^{3})^{3}}$ = $a ^{2}\left \| b| ^{3}$

5) $\sqrt[6]{a ^{4}-8a^{2}b ^{2}+ 16b^{4}}$ = $\sqrt[6]{(a ^{2}-4b^{2})^{2}}$ = $\sqrt[3]{a ^{2}-4b^{2}}$

6) $\sqrt[4]{a ^{2}+2ab +b ^{2}}$ = $\sqrt[4]{(a +b) ^{2}}$ = $\sqrt[2]{\left \| (a +b) |}$ in pratica il valore assoluto lo mettiamo ogni qual volta il radicale è pari è quando i componenti del radicando hanno un esponente dispari.

7) $\sqrt[6]{\frac{8a ^{3}+ 24a ^{2}b+24a b ^{2}+ 8b ^{3}}{(27a ^{3}-81a^{2}b +81a b ^{2}-27b ^{3})(x ^{3}+ 3x ^{2}y +3xy ^{2}+y^{3})}}$

$\sqrt[6]{\frac{8a ^{3}+ 24a ^{2}b+24a b ^{2}+ 8b ^{3}}{(27a ^{3}-81a^{2}b +81a b ^{2}-27b ^{3})(x ^{3}+ 3x ^{2}y +3xy ^{2}+y^{3})}}$ = $\sqrt[6]{\frac{(2a +2b) ^{3}}{(3a -3b) ^{3}(x +y) ^{3}}}$ = $\sqrt[2]{\frac{2a +2b}{(3a -3b) (x +y) }}$ con a>0 e x≠y

8) $\sqrt[4]{\frac{c ^{4}a ^{2}(x -2z)^{2}}{4(a -b)^{2}}}$

$\sqrt[4]{\frac{c ^{4}a ^{2}(x -2z)^{2}}{4(a -b)^{2}}}$ = $\sqrt[4]{\frac{c ^{4}a ^{2}(x -2z)^{2}}{2^{2}(a -b)^{2}}}$ = $\sqrt[2]{\frac{c ^{2}\left \| a| \left \| x-2z|}{2 \left \| a-b|}}$