Add and Subtract Square Roots

By the end of this section, you will be able to: add and subtract like square roots, and add and subtract square roots that need simplification.

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. That is why we simplify $\sqrt{2 + 7}$ by adding inside the radical to get $\sqrt{9}$ , which is $3$ .

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

\sqrt{2} + \sqrt{7} \neq \sqrt{2 + 7}

Trying to add square roots with different radicands is like trying to add unlike terms. But just like we can add $x + x$ to get $2x$ , we can add $\sqrt{3} + \sqrt{3}$ to get $2\sqrt{3}$ .

Add and subtract like square roots

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.

Like Square Roots. Square roots with the same radicand are called like square roots.

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 $11\sqrt{x}$ .

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.

Example. Simplify $2\sqrt{2} - 7\sqrt{2}$ .

The radicals are like, so we subtract the coefficients:

\begin{array}{lrcl} \text{Subtract the coefficients.} & 2\sqrt{2} - 7\sqrt{2} &=& -5\sqrt{2} \end{array}

Simplify: $8\sqrt{2} - 9\sqrt{2}$ .

Example. Simplify $3\sqrt{y} + 4\sqrt{y}$ .

The radicals are like, so we add the coefficients:

\begin{array}{lrcl} \text{Add the coefficients.} & 3\sqrt{y} + 4\sqrt{y} &=& 7\sqrt{y} \end{array}

Example. Simplify $4\sqrt{x} - 2\sqrt{y}$ .

The radicals are not like, so we cannot subtract them. We leave the expression as is:

4\sqrt{x} - 2\sqrt{y}

Simplify: $2\sqrt{x} + 7\sqrt{x}$ .

When there are three or more radicals, combine the like ones the same way.

Example. Simplify $5\sqrt{13} + 4\sqrt{13} + 2\sqrt{13}$ .

The radicals are like, so we add the coefficients:

\begin{array}{lrcl} \text{Add the coefficients.} & 5\sqrt{13} + 4\sqrt{13} + 2\sqrt{13} &=& 11\sqrt{13} \end{array}

Example. Simplify $2\sqrt{6} - 6\sqrt{6} + 3\sqrt{3}$ .

The first two radicals are like, so we subtract their coefficients. The $3\sqrt{3}$ term is not like the others, so it stays as it is:

\begin{array}{lrcl} \text{Combine the like radicals.} & 2\sqrt{6} - 6\sqrt{6} + 3\sqrt{3} &=& -4\sqrt{6} + 3\sqrt{3} \end{array}

Sometimes combining the coefficients gives $0$ , which makes the whole radical term $0$ .

Example. Simplify $2\sqrt{5n} - 6\sqrt{5n} + 4\sqrt{5n}$ .

The radicals are all like, so we combine them by combining the coefficients:

\begin{array}{lrcl} \text{Combine the like radicals.} & 2\sqrt{5n} - 6\sqrt{5n} + 4\sqrt{5n} &=& 0\sqrt{5n} \\[4pt] \text{Simplify.} & &=& 0 \end{array}

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

Example. Simplify $\sqrt{3xy} + 5\sqrt{3xy} - 4\sqrt{3xy}$ .

The radicals are all like, so we combine them by combining the coefficients ( $1 + 5 - 4$ ):

\begin{array}{lrcl} \text{Combine the like radicals.} & \sqrt{3xy} + 5\sqrt{3xy} - 4\sqrt{3xy} &=& 2\sqrt{3xy} \end{array}

Simplify: $\sqrt{5xy} + 4\sqrt{5xy} - 7\sqrt{5xy}$ .

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.

Example. Simplify $\sqrt{20} + 3\sqrt{5}$ .

Simplify the radicals when possible, then combine the like radicals:

\begin{array}{lrcl} \text{Simplify the radicals.} & \sqrt{20} + 3\sqrt{5} &=& \sqrt{4} \cdot \sqrt{5} + 3\sqrt{5} \\[4pt] & &=& 2\sqrt{5} + 3\sqrt{5} \\[4pt] \text{Combine the like radicals.} & &=& 5\sqrt{5} \end{array}

Example. Simplify $\sqrt{48} - \sqrt{75}$ .

Simplify each radical using the Product Property, then combine the like radicals:

\begin{array}{lrcl} \text{Simplify the radicals.} & \sqrt{48} - \sqrt{75} &=& \sqrt{16} \cdot \sqrt{3} - \sqrt{25} \cdot \sqrt{3} \\[4pt] & &=& 4\sqrt{3} - 5\sqrt{3} \\[4pt] \text{Combine the like radicals.} & &=& -\sqrt{3} \end{array}

Simplify: $\sqrt{32} - \sqrt{18}$ .

Just like we use the Associative Property of Multiplication to simplify $5(3x)$ and get $15x$ , we can simplify $5\left(3\sqrt{x}\right)$ and get $15\sqrt{x}$ . We use the Associative Property to do this in the next example.

Example. Simplify $5\sqrt{18} - 2\sqrt{8}$ .

Simplify the radicals, multiply the coefficients, then combine the like radicals:

\begin{array}{lrcl} \text{Simplify the radicals.} & 5\sqrt{18} - 2\sqrt{8} &=& 5 \cdot \sqrt{9} \cdot \sqrt{2} - 2 \cdot \sqrt{4} \cdot \sqrt{2} \\[4pt] & &=& 5 \cdot 3 \cdot \sqrt{2} - 2 \cdot 2 \cdot \sqrt{2} \\[4pt] & &=& 15\sqrt{2} - 4\sqrt{2} \\[4pt] \text{Combine the like radicals.} & &=& 11\sqrt{2} \end{array}

Simplify: $4\sqrt{27} - 3\sqrt{12}$ .

When the coefficients are fractions, simplify the radicals first, then find a common denominator to combine the coefficients of the like radicals.

Example. Simplify $\tfrac{3}{4}\sqrt{192} - \tfrac{5}{6}\sqrt{108}$ .

Simplify each radical, multiply the coefficients, then combine the like radicals:

\begin{array}{lrcl} \text{Simplify the radicals.} & \tfrac{3}{4}\sqrt{192} - \tfrac{5}{6}\sqrt{108} &=& \tfrac{3}{4}\sqrt{64} \cdot \sqrt{3} - \tfrac{5}{6}\sqrt{36} \cdot \sqrt{3} \\[4pt] & &=& \tfrac{3}{4} \cdot 8 \cdot \sqrt{3} - \tfrac{5}{6} \cdot 6 \cdot \sqrt{3} \\[4pt] & &=& 6\sqrt{3} - 5\sqrt{3} \\[4pt] \text{Combine the like radicals.} & &=& \sqrt{3} \end{array}

Example. Simplify $\tfrac{2}{3}\sqrt{48} - \tfrac{3}{4}\sqrt{12}$ .

Here the coefficients of the like radicals are fractions, so we find a common denominator before subtracting:

\begin{array}{lrcl} \text{Simplify the radicals.} & \tfrac{2}{3}\sqrt{48} - \tfrac{3}{4}\sqrt{12} &=& \tfrac{2}{3}\sqrt{16} \cdot \sqrt{3} - \tfrac{3}{4}\sqrt{4} \cdot \sqrt{3} \\[4pt] & &=& \tfrac{2}{3} \cdot 4 \cdot \sqrt{3} - \tfrac{3}{4} \cdot 2 \cdot \sqrt{3} \\[4pt] & &=& \tfrac{8}{3}\sqrt{3} - \tfrac{3}{2}\sqrt{3} \\[4pt] \text{Find a common denominator.} & &=& \tfrac{16}{6}\sqrt{3} - \tfrac{9}{6}\sqrt{3} \\[4pt] \text{Simplify.} & &=& \tfrac{7}{6}\sqrt{3} \end{array}

In the next example, we will remove constant and variable factors from the square roots.

Example. Simplify $\sqrt{18n^5} - \sqrt{32n^5}$ .

Simplify each radical by removing the largest perfect-square factor, then combine the like radicals:

\begin{array}{lrcl} \text{Simplify the radicals.} & \sqrt{18n^5} - \sqrt{32n^5} &=& \sqrt{9n^4} \cdot \sqrt{2n} - \sqrt{16n^4} \cdot \sqrt{2n} \\[4pt] & &=& 3n^2\sqrt{2n} - 4n^2\sqrt{2n} \\[4pt] \text{Combine the like radicals.} & &=& -n^2\sqrt{2n} \end{array}

Sometimes, even after simplifying, the radicals are still not like and cannot be combined.

Example. Simplify $9\sqrt{50m^2} - 6\sqrt{48m^2}$ .

Simplify each radical, multiply the coefficients, then check whether the radicals are like:

\begin{array}{lrcl} \text{Simplify the radicals.} & 9\sqrt{50m^2} - 6\sqrt{48m^2} &=& 9\sqrt{25m^2} \cdot \sqrt{2} - 6\sqrt{16m^2} \cdot \sqrt{3} \\[4pt] & &=& 9 \cdot 5m \cdot \sqrt{2} - 6 \cdot 4m \cdot \sqrt{3} \\[4pt] & &=& 45m\sqrt{2} - 24m\sqrt{3} \end{array}

The radicals $\sqrt{2}$ and $\sqrt{3}$ are not like, so the expression cannot be combined any further.

Example. Simplify $2\sqrt{8x^2} - 5x\sqrt{32} + 5\sqrt{18x^2}$ .

Simplify each radical, multiply the coefficients, then combine the like radicals (all three simplify to a multiple of $\sqrt{2}$ ):

\begin{array}{lrcl} \text{Simplify the radicals.} & 2\sqrt{8x^2} - 5x\sqrt{32} + 5\sqrt{18x^2} &=& 2\sqrt{4x^2} \cdot \sqrt{2} - 5x\sqrt{16} \cdot \sqrt{2} + 5\sqrt{9x^2} \cdot \sqrt{2} \\[4pt] & &=& 2 \cdot 2x \cdot \sqrt{2} - 5x \cdot 4 \cdot \sqrt{2} + 5 \cdot 3x \cdot \sqrt{2} \\[4pt] & &=& 4x\sqrt{2} - 20x\sqrt{2} + 15x\sqrt{2} \\[4pt] \text{Combine the like radicals.} & &=& -x\sqrt{2} \end{array}

Simplify: $\tfrac{2}{5}\sqrt{32} - \tfrac{1}{3}\sqrt{8}$ .

Simplify: $\sqrt{27p^3} - \sqrt{48p^3}$ .

Key terms

like square roots — square roots with the same radicand (such as $3\sqrt{x}$ and $8\sqrt{x}$ ); like square roots can be combined by adding or subtracting their coefficients, just like like terms. coefficient — the numerical factor multiplying a square root; only the coefficients of like square roots are combined when adding or subtracting.

This section is adapted from Elementary Algebra 2e, 9.3 Add and Subtract Square Roots by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked examples as prose with aligned display step tables, folded the order-of-operations warm-up into the section opener, and stated the like-square-roots definition inline; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Simplify Square Roots Multiply Square Roots