9.3: Add and Subtract Square Roots

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Learning Objectives

By the end of this section, you will be able to:

Add and subtract like square roots
Add and subtract square roots that need simplification

BE PREPARED

Before you get started, take this readiness quiz.

Add: ⓐ \(3x+9x\) ⓑ \(5m+5n\).
If you missed this problem, review [link].
Simplify: \(\sqrt{50x^3}\).
If you missed this problem, review [link].

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:

\[\begin{array}{ll} {}&{\sqrt{2+7}}\\ {\text{Add inside the radical.}}&{\sqrt{9}}\\ {\text{Simplify.}}&{3}\\ \end{array}\]

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

\(\sqrt{2}+\sqrt{7} \ne \sqrt{2+7}\)

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

\[\begin{array}{llll} {\text{But, just like we can}}&{x+x}&{\text{we can add}}&{\sqrt{3}+\sqrt{3}}\\ {}&{x+x=2x}&{}&{\sqrt{3}+\sqrt{3}=2\sqrt{3}}\\ \end{array}\]

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.

Definition: 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 \(11\sqrt{x}\).

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.

Example \(\PageIndex{1}\)

Simplify: \(2\sqrt{2}−7\sqrt{2}\).

Answer: \[\begin{array}{ll} {}&{2\sqrt{2}−7\sqrt{2}}\\ {\text{Since the radicals are like, we subtract the coefficients.}}&{−5\sqrt{2}}\\ \end{array}\]

Example \(\PageIndex{2}\)

Simplify: \(8\sqrt{2}−9\sqrt{2}\).

Answer: \(−\sqrt{2}\)

Example \(\PageIndex{3}\)

Simplify: \(5\sqrt{3}−9\sqrt{3}\).

Answer: \(−4\sqrt{3}\)

Example \(\PageIndex{4}\)

Simplify: \(3\sqrt{y}+4\sqrt{y}\).

Answer: \[\begin{array}{ll} {}&{3\sqrt{y}+4\sqrt{y}}\\ {\text{Since the radicals are like, we add the coefficients.}}&{7\sqrt{y}}\\ \end{array}\]

Example \(\PageIndex{5}\)

Simplify: \(2\sqrt{x}+7\sqrt{x}\).

Answer: \(9\sqrt{x}\)

Example \(\PageIndex{6}\)

Simplify: \(5\sqrt{u}+3\sqrt{u}\).

Answer: \(8\sqrt{u}\)

Example \(\PageIndex{7}\)

Simplify: \(4\sqrt{x}−2\sqrt{y}\)

Answer: \[\begin{array}{ll} {}&{4\sqrt{x}−2\sqrt{y}}\\ {\text{Since the radicals are not like, we cannot subtract them. We leave the expression as is.}}&{4\sqrt{x}−2\sqrt{y}}\\ \end{array}\]

Example \(\PageIndex{8}\)

Simplify: \(7\sqrt{p}−6\sqrt{q}\).

Answer: \(7\sqrt{p}−6\sqrt{q}\)

Example \(\PageIndex{9}\)

Simplify: \(6\sqrt{a}−3\sqrt{b}\).

Answer: \(6\sqrt{a}−3\sqrt{b}\)

Example \(\PageIndex{10}\)

Simplify: \(5\sqrt{13}+4\sqrt{13}+2\sqrt{13}\).

Answer: \[\begin{array}{ll} {}&{5\sqrt{13}+4\sqrt{13}+2\sqrt{13}}\\ {\text{Since the radicals are like, we add the coefficients.}}&{11\sqrt{13}}\\ \end{array}\]

Example \(\PageIndex{11}\)

Simplify: \(4\sqrt{11}+2\sqrt{11}+3\sqrt{11}\).

Answer: \(9\sqrt{11}\)

Example \(\PageIndex{12}\)

Simplify: \(6\sqrt{10}+2\sqrt{10}+3\sqrt{10}\).

Answer: \(11\sqrt{10}\)

Example \(\PageIndex{13}\)

Simplify: \(2\sqrt{6}−6\sqrt{6}+3\sqrt{3}\).

Answer: \[\begin{array}{ll} {}&{2\sqrt{6}−6\sqrt{6}+3\sqrt{3}}\\ {\text{Since the first two radicals are like, we subtract their coefficients.}}&{−4\sqrt{6}+3\sqrt{3}}\\ \end{array}\]

Example \(\PageIndex{14}\)

Simplify: \(5\sqrt{5}−4\sqrt{5}+2\sqrt{6}\).

Answer: \(\sqrt{5}+2\sqrt{6}\)

Example \(\PageIndex{15}\)

Simplify: \(3\sqrt{7}−8\sqrt{7}+2\sqrt{5}\).

Answer: \(−5\sqrt{7}+2\sqrt{5}\)

Example \(\PageIndex{16}\)

Simplify: \(2\sqrt{5n}−6\sqrt{5n}+4\sqrt{5n}\).

Answer: \[\begin{array}{ll} {}&{2\sqrt{5n}−6\sqrt{5n}+4\sqrt{5n}}\\ {\text{Since the radicals are like, we combine them.}}&{−0\sqrt{5n}}\\ {\text{Simplify.}}&{0}\\ \end{array}\]

Example \(\PageIndex{17}\)

Simplify: \(\sqrt{7x}−7\sqrt{7x}+4\sqrt{7x}\).

Answer: \(−2\sqrt{7x}\)

Example \(\PageIndex{18}\)

Simplify: \(4\sqrt{3y}−7\sqrt{3y}+2\sqrt{3y}\).

Answer: \(−3\sqrt{y}\)

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

Example \(\PageIndex{19}\)

Simplify: \(\sqrt{3xy}+5\sqrt{3xy}−4\sqrt{3xy}\).

Answer: \[\begin{array}{ll} {}&{\sqrt{3xy}+5\sqrt{3xy}−4\sqrt{3xy}}\\ {\text{Since the radicals are like, we combine them.}}&{2\sqrt{3xy}}\\ \end{array}\]

Example \(\PageIndex{20}\)

Simplify: \(\sqrt{5xy}+4\sqrt{5xy}−7\sqrt{5xy}\).

Answer: \(−2\sqrt{5xy}\)

Example \(\PageIndex{21}\)

Simplify: \(3\sqrt{7mn}+\sqrt{7mn}−4\sqrt{7mn}\).

Answer: 0

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 \(\PageIndex{22}\)

Simplify: \(\sqrt{20}+3\sqrt{5}\).

Answer: \[\begin{array}{ll} {}&{\sqrt{20}+3\sqrt{5}}\\ {\text{Simplify the radicals, when possible.}}&{\sqrt{4}·\sqrt{5}+3\sqrt{5}}\\ {}&{2\sqrt{5}+3\sqrt{5}}\\ {\text{Combine the like radicals.}}&{5\sqrt{5}}\\ \end{array}\]

Example \(\PageIndex{23}\)

Simplify: \(\sqrt{18}+6\sqrt{2}\).

Answer: \(9\sqrt{2}\)

Example \(\PageIndex{24}\)

Simplify: \(\sqrt{27}+4\sqrt{3}\).

Answer: \(7\sqrt{3}\)

Example \(\PageIndex{25}\)

Simplify: \(\sqrt{48}−\sqrt{75}\)

Answer: \[\begin{array}{ll} {}&{\sqrt{48}−\sqrt{75}}\\ {\text{Simplify the radicals.}}&{\sqrt{16}·\sqrt{3}−\sqrt{25}·\sqrt{3}}\\ {}&{4\sqrt{3}−5\sqrt{3}}\\ {\text{Combine the like radicals.}}&{−\sqrt{3}}\\ \end{array}\]

Example \(\PageIndex{26}\)

Simplify: \(\sqrt{32}−\sqrt{18}\).

Answer: \(\sqrt{2}\)

Example \(\PageIndex{27}\)

Simplify: \(\sqrt{20}−\sqrt{45}\).

Answer: \(−\sqrt{5}\)

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

Example \(\PageIndex{28}\)

Simplify: \(5\sqrt{18}−2\sqrt{8}\).

Answer: \[\begin{array}{ll} {}&{5\sqrt{18}−2\sqrt{8}}\\ {\text{Simplify the radicals.}}&{5·\sqrt{9}·\sqrt{2}−2·\sqrt{4}·\sqrt{2}}\\ {}&{5·3·\sqrt{2}−2·2·\sqrt{2}}\\ {}&{15\sqrt{2}−4\sqrt{2}}\\ {\text{Combine the like radicals.}}&{11\sqrt{2}}\\ \end{array}\]

Example \(\PageIndex{29}\)

Simplify: \(4\sqrt{27}−3\sqrt{12}\).

Answer: \(6\sqrt{3}\)

Example \(\PageIndex{30}\)

Simplify: \(3\sqrt{20}−7\sqrt{45}\).

Answer: \(−15\sqrt{5}\)

Example \(\PageIndex{31}\)

Simplify: \(\frac{3}{4}\sqrt{192}−\frac{5}{6}\sqrt{108}\).

Answer: \[\begin{array}{ll} {}&{\frac{3}{4}\sqrt{192}−\frac{5}{6}\sqrt{108}}\\ {\text{Simplify the radicals.}}&{\frac{3}{4}\sqrt{64}·\sqrt{3}−\frac{5}{6}\sqrt{36}·\sqrt{3}}\\ {}&{\frac{3}{4}·8·\sqrt{3}−\frac{5}{6}·6·\sqrt{3}}\\ {}&{6\sqrt{3}−5\sqrt{3}}\\ {\text{Combine the like radicals.}}&{\sqrt{3}}\\ \end{array}\]

Example \(\PageIndex{32}\)

Simplify: \(\frac{2}{3}\sqrt{108}−\frac{5}{7}\sqrt{147}\).

Answer: \(−\sqrt{3}\)

Example \(\PageIndex{33}\)

Simplify: \(\frac{3}{5}\sqrt{200}−\frac{3}{4}\sqrt{128}\).

Answer: 0

Example \(\PageIndex{34}\)

Simplify: \(\frac{2}{3}\sqrt{48}−\frac{3}{4}\sqrt{12}\).

Answer: \[\begin{array}{ll} {}&{\frac{2}{3}\sqrt{48}−\frac{3}{4}\sqrt{12}}\\ {\text{Simplify the radicals.}}&{\frac{2}{3}\sqrt{16}·\sqrt{3}−\frac{3}{4}\sqrt{4}·\sqrt{3}}\\ {}&{\frac{2}{3}·4·\sqrt{3}−\frac{3}{4}·2·\sqrt{3}}\\ {}&{\frac{8}{3}\sqrt{3}−\frac{3}{2}\sqrt{3}}\\ {\text{Find a common denominator to subtract the coefficients of the like radicals.}}&{\frac{16}{6}\sqrt{3}−\frac{9}{6}\sqrt{3}}\\ {\text{Simplify.}}&{\frac{7}{6}\sqrt{3}} \end{array}\]

Example \(\PageIndex{35}\)

Simplify: \(\frac{2}{5}\sqrt{32}−\frac{1}{3}\sqrt{8}\)

Answer: \(\frac{14}{15}\sqrt{2}\)

Example \(\PageIndex{36}\)

Simplify: \(\frac{1}{3}\sqrt{80}−\frac{1}{4}\sqrt{125}\)

Answer: \(\frac{1}{12}[\sqrt{5}\)

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

Example \(\PageIndex{37}\)

Simplify: \(\sqrt{18n^5}−\sqrt{32n^5}\)

Answer: \[\begin{array}{ll} {}&{\sqrt{18n^5}−\sqrt{32n^5}}\\ {\text{Simplify the radicals.}}&{\sqrt{9n^4}·\sqrt{2n}−\sqrt{16n^4}·\sqrt{2n}}\\ {}&{3n^2\sqrt{2n}−4n^2\sqrt{2n}}\\ {\text{Combine the like radicals.}}&{−n^2\sqrt{2n}}\\ \end{array}\]

Example \(\PageIndex{38}\)

Simplify: \(\sqrt{32m^7}−\sqrt{50m^7}\).

Answer: \(−m^3\sqrt{2m}\)

Example \(\PageIndex{39}\)

Simplify: \(\sqrt{27p^3}−\sqrt{48p^3}\)

Answer: \(−p^3\sqrt{p}\)

Example \(\PageIndex{40}\)

Simplify: \(9\sqrt{50m^2}−6\sqrt{48m^2}\).

Answer: \[\begin{array}{ll} {}&{9\sqrt{50m^{2}}−6\sqrt{48m^{2}}}\\ {\text{Simplify the radicals.}}&{9\sqrt{25m^{2}}·\sqrt{2}−6·\sqrt{16m^{2}}·\sqrt{3}}\\ {}&{9·5m·\sqrt{2}−6·4m·\sqrt{3}}\\ {}&{45m\sqrt{2}−24m\sqrt{3}}\\ \end{array}\]

Example \(\PageIndex{41}\)

Simplify: \(5\sqrt{32x^2}−3\sqrt{48x^2}\).

Answer: \(20x\sqrt{2}−12x\sqrt{3}\)

Example \(\PageIndex{42}\)

Simplify: \(7\sqrt{48y^2}−4\sqrt{72y^2}\).

Answer: \(28y\sqrt{3}−24y\sqrt{2}\)

Example \(\PageIndex{43}\)

Simplify: \(2\sqrt{8x^2}−5x\sqrt{32}+5\sqrt{18x^2}\).

Answer: \[\begin{array}{ll} {}&{2\sqrt{8x^2}−5x\sqrt{32}+5\sqrt{18x^2}}\\ {\text{Simplify the radicals.}}&{2\sqrt{4x^2}·\sqrt{2}−5x\sqrt{16}·\sqrt{2}+5\sqrt{9x^2}·\sqrt{2}}\\ {}&{2·2x·\sqrt{2}−5x·4·\sqrt{2}+5·3x·\sqrt{2}}\\ {}&{4x\sqrt{2}−20x\sqrt{2}+15x\sqrt{2}}\\ {\text{Combine the like radicals.}}&{−x\sqrt{2}}\\ \end{array}\]

Example \(\PageIndex{44}\)

Simplify: \(3\sqrt{12x^2}−2x\sqrt{48}+4\sqrt{27x^2}\)

Answer: \(10x\sqrt{3}\)

Example \(\PageIndex{45}\)

Simplify: \(3\sqrt{18x^2}−6x\sqrt{32}+2\sqrt{50x^2}\).

Answer: \(−5x\sqrt{2}\)

Access this online resource for additional instruction and practice with the adding and subtracting square roots.

Adding/Subtracting Square Roots

Glossary

like square roots: Square roots with the same radicand are called like square roots.