$\sqrt{2}·\sqrt{18}$
= $\sqrt{2}·\sqrt{2·9}$
= $\sqrt{2}·\sqrt{2}·\sqrt{9}$
= $2·3$
= $6$

$\sqrt{2}·\sqrt{\frac{1}{2}}$
= $\sqrt{2·\frac{1}{2}}$
= $\sqrt{1·1}$
= $\sqrt{1}$
= $1$

$\left(\sqrt{18}+\sqrt{2}\right)·\sqrt{2}$
= $\sqrt{18}·\sqrt{2}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2·9}·\sqrt{2}+2$
= $\sqrt{9}·\sqrt{2}·\sqrt{2}+2$
= $3·2+2$
= $6+2$
= $8$

Nach der 3. binomischen Formel ((a+b)(a-b)=a²-b²) gilt:

$\left(\sqrt{8}-7\right)·\left(\sqrt{8}+7\right)$ = ${\left(\sqrt{8}\right)}^2-7^2$
= $8-49$
= $-41$

$5\sqrt{x}+\sqrt{64x}$
= $5\sqrt{x}+\sqrt{64}·\sqrt{x}$
= $5\sqrt{x}+8\sqrt{x}$
= $13\sqrt{x}$

Wir können die $\sqrt{21}$ wie Einheiten behandeln und einfach verrechnen.

$14\sqrt{21}+6\sqrt{21}$
= $20\sqrt{21}$