# $$\left[\frac{3 \sqrt{12} – 3 \sqrt{2}}{\sqrt{18}} + \sqrt[4]{(\frac{3}{2})^2} · (\sqrt{6} – 2)\right] · \frac{1}{(\sqrt{2} – 2)^2}$$

Semplifica la seguente espressione numerica ed esprimi il risultato in modo che gli eventuali denominatori non contengano radicali.

$$\left[\frac{3 \sqrt{12} – 3 \sqrt{2}}{\sqrt{18}} + \sqrt[4]{(\frac{3}{2})^2} · (\sqrt{6} – 2)\right] · \frac{1}{(\sqrt{2} – 2)^2}$$

### Svolgimento

$$\left[ \frac{3 \sqrt{3 · 4} – 3 \sqrt{2}}{\sqrt{2 · 9}} + \sqrt[4]{ \left(\frac{3}{2} \right)^2} · (\sqrt{6} – 2) \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$\left[\frac{3 · 2 · \sqrt{3} – 3 \sqrt{2}}{3 \sqrt{2}} + \sqrt[4]{ \left(\frac{3}{2}\right)^2} · (\sqrt{6} – 2) \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$\left[\frac{ 6 \sqrt{3} – 3 \sqrt{2}}{3 \sqrt{2}} + \sqrt[4]{ \left(\frac{3}{2}\right)^2} · (\sqrt{6} – 2) \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$\left[\frac{ 6 \sqrt{3} – 3 \sqrt{2}}{3 \sqrt{2}} + \sqrt[2]{\frac{3}{2}} · (\sqrt{6} – 2) \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

Svolgiamo la moltiplicazione:

$$\left[\frac{ 6 \sqrt{3} – 3 \sqrt{2}}{3 \sqrt{2}} + \sqrt{\frac{3}{2}} · \sqrt{6} – \sqrt{\frac{3}{2}} · 2 \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$\left[\frac{ 6 \sqrt{3} – 3 \sqrt{2}}{3 \sqrt{2}} + \sqrt{\frac{3}{2} · 6} – 2 \sqrt{\frac{3}{2}} \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$\left[\frac{ 6 \sqrt{3} – 3 \sqrt{2}}{3 \sqrt{2}} + 3 – 2 \sqrt{\frac{3}{2}} \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

Calcoliamo il minimo comune multiplo all’interno della parentesi:

$$\left[\frac{ 6 \sqrt{3} – 3 \sqrt{2} + 9 \sqrt{2} – 6 \sqrt{3}}{3 \sqrt{2}} \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$\left[\frac{ 6 \sqrt{2}}{3 \sqrt{2}} \right] · \frac{1}{(\sqrt{2} – 2)^2}$$

$$2 · \frac{1}{(\sqrt{2} – 2)^2}$$

$2 · frac(1)(2 + 4 – 4sqrt2)$

$2 · frac(1)(6 – 4sqrt2)$

Mettiamo in evidenza e semplifichiamo:

$2 · frac(1)(2(3 – 2sqrt2)) = frac(1)(3 – 2sqrt2)$

Razionalizziamo:

$frac(1)(3 – 2sqrt2) · frac(3 + 2 sqrt2)(3 + 2 sqrt2) = frac(3 + 2 sqrt2)((3 – 2 sqrt2)(3 + 2 sqrt2)) =$

$frac(3 + 2 sqrt2)(9 – 8) = 3 + 2 sqrt2$

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