Exercises: Computing with Square Roots

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

Use the root law for adding roots.

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

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

Use the root law for subtracting roots and simplify.

$\begin{array}{l}=(6-2)\sqrt{4}\\ =4\sqrt{4}=4\cdot 2=8\end{array}$

Alternative solution

You can also calculate the solution without the root laws:

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

Calculate the root.

$6\cdot 2-2\cdot 2=$

Calculate.

$12-4=8$

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

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: $3\cdot (\sqrt{4}+\sqrt{3})$

However, this is not always helpful.

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

Use the root law for multiplying roots.

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

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

Use the root law for multiplying roots.

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

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

Use the root law for dividing roots.

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

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

Use the root law for taking roots of squares.

$=|-27|=27$

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

Use the root law for taking squares of a root.

$=18$

$\sqrt{7}\cdot \sqrt{7}$

Use the root law for multiplying roots.

$=7$

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

Use the root law for converting powers into roots.

$=\sqrt{25}=5$