9.3 Add and subtract square roots

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify $\sqrt{2+7}$ in this way:

$\begin{array}{cccc}& & & \sqrt{2+7}\hfill \\ \text{Add inside the radical.}\hfill & & & \sqrt{9}\hfill \\ \text{Simplify.}\hfill & & & 3\hfill \end{array}$

So if we have to add $\sqrt{2}+\sqrt{7}$ , we must not combine them into one radical.

Trying to add square roots with different radicands is like trying to add unlike terms.

$\begin{array}{cccc}\text{But, just like we can add}\hfill & x+x,\hfill & \text{we can add}\hfill & \sqrt{3}+\sqrt{3}.\hfill \\ & x+x=2x\hfill & & \sqrt{3}+\sqrt{3}=2\sqrt{3}\hfill \end{array}$

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

We add and subtract like square roots in the same way we add and subtract like terms. We know that $3x+8x$ is $11x$ . Similarly we add $3\sqrt{x}+8\sqrt{x}$ and the result is $11\sqrt{x}.$

Add and subtract like square roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.

When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

Add and subtract square roots that need simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals , we find like radicals after simplifying the square roots.