$5\sqrt{3}+2\sqrt{3}=5\; \backslash sqrt\{3\}\; \backslash ;+2\backslash sqrt\{3\}=$

Use the root law for adding roots.

$(5+2)\sqrt{3}=7\sqrt{3}(5+2)\backslash sqrt\{3\}=7\backslash sqrt\{3\}$

$6\sqrt{4}-2\sqrt{4}=6\; \backslash sqrt\{4\}\; \backslash ;-2\backslash sqrt\{4\}=$

Use the root law for subtracting roots and simplify.

$=(6-2)\sqrt{4}=4\sqrt{4}=4\cdot 2=8=(6-2)\; \backslash sqrt\{4\}\backslash \backslash \; =4\; \backslash sqrt\{4\}=4\; \backslash cdot\; 2\; =\; 8$

Alternative solution

You can also calculate the solution without the root laws:

$6\sqrt{4}-2\sqrt{4}=6\; \backslash sqrt\{4\}\; \backslash ;-2\backslash sqrt\{4\}=$

Calculate the root.

$6\cdot 2-2\cdot 2=6\; \backslash cdot\; 2\; \backslash ;-2\backslash cdot\; 2=$

Calculate.

$12-4=812-4=8$

$3\sqrt{4}+3\sqrt{3}3\backslash sqrt\{4\}+3\backslash sqrt\{3\}\backslash ;$

Here you cannot use the root law to add roots because the radicands are not equal. So you can't further simplify this term.

What you could do is factoring out a three: $3\cdot (\sqrt{4}+\sqrt{3})3\backslash cdot(\backslash sqrt\{4\}+\backslash sqrt\{3\})\backslash ;$

However, this is not always helpful.

$\sqrt{3}\cdot \sqrt{7}=\backslash sqrt\{3\}\; \backslash cdot\; \backslash sqrt\{7\}=$

Use the root law for multiplying roots.

$=\sqrt{3\cdot 7}=\sqrt{21}=\backslash sqrt\{3\; \backslash cdot\; 7\}\; =\; \backslash sqrt\{21\}$

$\sqrt{2}\cdot \sqrt{8}=\backslash sqrt\{2\}\backslash cdot\; \backslash sqrt\{8\}=$

Use the root law for multiplying roots.

$\sqrt{2\cdot 8}=\sqrt{16}=4\backslash sqrt\{2\; \backslash cdot\; 8\}=\backslash sqrt\{16\}=4$

$\frac{\sqrt{27}}{\sqrt{3}}=\backslash frac\{\backslash sqrt\{27\}\}\{\backslash sqrt\{3\}\}=$

Use the root law for dividing roots.

$=\sqrt{\frac{27}{3}}=\sqrt{9}=3=\backslash sqrt\{\backslash frac\{27\}\; \{3\}\}=\backslash sqrt\; \{9\}=3$

$\sqrt{(-27{)}^2}\backslash sqrt\{(-27)^2\}$

Use the root law for taking roots of squares.

$=\mathrm{\mid }-27\mathrm{\mid }=27=|-27|=27$

$(\sqrt{2\cdot 9}{)}^2=(\sqrt{18}{)}^2(\backslash sqrt\{2\backslash cdot9\})^2=(\backslash sqrt\{18\})^2$

Use the root law for taking squares of a root.

$=18=18$

$\sqrt{7}\cdot \sqrt{7}\backslash sqrt7\backslash cdot\backslash sqrt7$

Use the root law for multiplying roots.

$=7=7$

${25}^{\frac{1}{2}}25^\{\backslash frac12\}$

Use the root law for converting powers into roots.

$=\sqrt{25}=5=\backslash sqrt\{25\}=5$