Adding and Subtracting Radicals

Radicals can be added or subtracted only if they are like radicals.

Like radicals are those having the same index or order.

 $Example:$   $-3\sqrt{2\ }\ and\ 5\ \sqrt{2}$  

If the radicals are unlike, we need to simplify.

Add the coefficients.

Simplify

 $Example\ :\ 4\sqrt{2}+2\sqrt{2}-\sqrt{2}$  

 $\left(4+2-1\right)\sqrt{2}\ \ \ \ \$  

 $5\sqrt{2}$  

Factor out to find a perfect root.

Simplify then operate.

Example of Factorization with Perfect root

 $2\sqrt{8}=2\sqrt{\left(4\right)\left(2\right)}=2\left(\sqrt{4}\right)\left(\sqrt{2}\right)=2\left(2\right)\left(\sqrt{2}\right)=4\sqrt{2}$  

 $5\sqrt{48}=5\ \sqrt{\left(16\right)\left(3\right)}=5\left(\sqrt{16}\right)\left(\sqrt{3}\right)=5\left(4\right)\left(\sqrt{3}\right)=20\sqrt{3}$  

 $5\sqrt{50}+3\sqrt{8}-8\sqrt{32}$                                 Factor out perfect roots.

 $5\sqrt{\left(25\right)\left(2\right)}+3\sqrt{\left(4\right)\left(2\right)}-8\sqrt{\left(16\right)\left(2\right)}$   Simplify perfect roots and multiply to coefficient.

 $5\left(5\right)\sqrt{2}+3\left(2\right)\sqrt{2}-8\left(4\right)\sqrt{2}$ 

 $25\sqrt{2}+6\sqrt{2}-24\sqrt{2}$                                  Operate 

 $\left(25+6-24\right)\sqrt{2}$                                         Simplify

 $7\sqrt{2}$