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:

$\begin{matrix} \sqrt{2 + 7} \\ Add inside the radical. & \sqrt{9} \\ Simplify. & 3 \end{matrix}$

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.

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

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 $3 x + 8 x$ is $11 x$ . 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.

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

Solution

Example demonstrating the subtraction of like radicals.
	$2 \sqrt{2} - 7 \sqrt{2}$
Since the radicals are like, we subtract the coefficients.	$−5 \sqrt{2}$

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

Solution

Demonstrates simplifying like radical expressions by adding their coefficients to obtain a combined radical term.
	$3 \sqrt{y} + 4 \sqrt{y}$
Since the radicals are like, we add the coefficients.	$7 \sqrt{y}$

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

Solution

This table demonstrates why expressions with unlike radicals cannot be subtracted, illustrating the final form of such an expression.
	$4 \sqrt{x} - 2 \sqrt{y}$
Since the radicals are not like, we cannot subtract them. We leave the expression as is.	$4 \sqrt{x} - 2 \sqrt{y}$

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

Solution

This table illustrates the process of adding like radicals, showing the initial expression, the rule applied, and the simplified result.
	$5 \sqrt{13} + 4 \sqrt{13} + 2 \sqrt{13}$
Since the radicals are like, we add the coefficients.	$11 \sqrt{13}$

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

Solution

This table illustrates the step-by-step simplification of a radical expression by combining like terms, showing the operation and the resulting expression.
	$2 \sqrt{6} - 6 \sqrt{6} + 3 \sqrt{3}$
Since the first two radicals are like, we subtract their coefficients.	$- 4 \sqrt{6} + 3 \sqrt{3}$

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

Solution

Step-by-step simplification of an algebraic expression involving like radicals, demonstrating its reduction to zero.
	$2 \sqrt{5 n} - 6 \sqrt{5 n} + 4 \sqrt{5 n}$
Since the radicals are like, we combine them.	$0 \sqrt{5 n}$
Simplify.	0

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

Simplify: $\sqrt{3 x y} + 5 \sqrt{3 x y} - 4 \sqrt{3 x y}$ .

Solution

This table demonstrates the step-by-step simplification of a mathematical expression involving like radicals.
	$\sqrt{3 x y} + 5 \sqrt{3 x y} - 4 \sqrt{3 x y}$
Since the radicals are like, we combine them.	$2 \sqrt{3 x y}$

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.

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

Solution

This table illustrates the step-by-step process of simplifying and combining radical expressions.
	$\sqrt{20} + 3 \sqrt{5}$
Simplify the radicals, when possible.	$\sqrt{4} \cdot \sqrt{5} + 3 \sqrt{5}$
	$2 \sqrt{5} + 3 \sqrt{5}$
Combine the like radicals.	$5 \sqrt{5}$

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

Solution

Steps to simplify and combine square root expressions.
	$\sqrt{48} - \sqrt{75}$
Simplify the radicals.	$\sqrt{16} \cdot \sqrt{3} - \sqrt{25} \cdot \sqrt{3}$
	$4 \sqrt{3} - 5 \sqrt{3}$
Combine the like radicals.	$− \sqrt{3}$

Just like we use the Associative Property of Multiplication to simplify $5 (3 x)$ and get $15 x$ , 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.

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

Solution

This table demonstrates the step-by-step simplification and combination of radical expressions.
	$5 \sqrt{18} - 2 \sqrt{8}$
Simplify the radicals.	$5 \cdot \sqrt{9} \cdot \sqrt{2} - 2 \cdot \sqrt{4} \cdot \sqrt{2}$
	$5 \cdot 3 \cdot \sqrt{2} - 2 \cdot 2 \cdot \sqrt{2}$
	$15 \sqrt{2} - 4 \sqrt{2}$
Combine the like radicals.	$11 \sqrt{2}$

Simplify: $\frac{3}{4} \sqrt{192} - \frac{5}{6} \sqrt{108}$ .

Solution

Step-by-step simplification of an algebraic expression involving radicals.
	$\frac{3}{4} \sqrt{192} - \frac{5}{6} \sqrt{108}$
Simplify the radicals.	$\frac{3}{4} \sqrt{64} \cdot \sqrt{3} - \frac{5}{6} \sqrt{36} \cdot \sqrt{3}$
	$\frac{3}{4} \cdot 8 \cdot \sqrt{3} - \frac{5}{6} \cdot 6 \cdot \sqrt{3}$
	$6 \sqrt{3} - 5 \sqrt{3}$
Combine the like radicals.	$\sqrt{3}$

Simplify: $\frac{2}{3} \sqrt{48} - \frac{3}{4} \sqrt{12}$ .

Solution

Step-by-step simplification of a radical mathematical expression.
	$\frac{2}{3} \sqrt{48} - \frac{3}{4} \sqrt{12}$
Simplify the radicals.	$\frac{2}{3} \sqrt{16} \cdot \sqrt{3} - \frac{3}{4} \sqrt{4} \cdot \sqrt{3}$
	$\frac{2}{3} \cdot 4 \cdot \sqrt{3} - \frac{3}{4} \cdot 2 \cdot \sqrt{3}$
	$\frac{8}{3} \sqrt{3} - \frac{3}{2} \sqrt{3}$
Find a common denominator to subtract the coefficients of the like radicals.	$\frac{16}{6} \sqrt{3} - \frac{9}{6} \sqrt{3}$
Simplify.	$\frac{7}{6} \sqrt{3}$

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

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

Solution

Illustrates the step-by-step simplification of an algebraic expression involving the subtraction of radical terms.
	$\sqrt{18 n^{5}} - \sqrt{32 n^{5}}$
Simplify the radicals.	$\sqrt{9 n^{4}} \cdot \sqrt{2 n} - \sqrt{16 n^{4}} \cdot \sqrt{2 n}$
	$3 n^{2} \sqrt{2 n} - 4 n^{2} \sqrt{2 n}$
Combine the like radicals.	$− n^{2} \sqrt{2 n}$

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

Solution

Step-by-step simplification of an algebraic expression with radicals, showing reduction and the condition for combining terms.
	$9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}}$
Simplify the radicals.	$9 \sqrt{25 m^{2}} \cdot \sqrt{2} - 6 \sqrt{16 m^{2}} \cdot \sqrt{3}$
	$9 \cdot 5 m \cdot \sqrt{2} - 6 \cdot 4 m \cdot \sqrt{3}$
	$45 m \sqrt{2} - 24 m \sqrt{3}$
The radicals are not like and so cannot be combined.

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

Solution

Demonstrates the step-by-step simplification of an algebraic expression containing radicals.
	$2 \sqrt{8 x^{2}} - 5 x \sqrt{32} + 5 \sqrt{18 x^{2}}$
Simplify the radicals.	$2 \sqrt{4 x^{2}} \cdot \sqrt{2} - 5 x \sqrt{16} \cdot \sqrt{2} + 5 \sqrt{9 x^{2}} \cdot \sqrt{2}$
	$2 \cdot 2 x \cdot \sqrt{2} - 5 x \cdot 4 \cdot \sqrt{2} + 5 \cdot 3 x \cdot \sqrt{2}$
	$4 x \sqrt{2} - 20 x \sqrt{2} + 15 x \sqrt{2}$
Combine the like radicals.	$− x \sqrt{2}$

Key Concepts

To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
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.

Practice Makes Perfect

Add and Subtract Like Square Roots

In the following exercises, simplify.

$8 \sqrt{2} - 5 \sqrt{2}$

Solution

$3 \sqrt{2}$

$7 \sqrt{2} - 3 \sqrt{2}$

$3 \sqrt{5} + 6 \sqrt{5}$

Solution

$9 \sqrt{5}$

$4 \sqrt{5} + 8 \sqrt{5}$

$9 \sqrt{7} - 10 \sqrt{7}$

Solution

$− \sqrt{7}$

$11 \sqrt{7} - 12 \sqrt{7}$

$7 \sqrt{y} + 2 \sqrt{y}$

Solution

$9 \sqrt{y}$

$9 \sqrt{n} + 3 \sqrt{n}$

$\sqrt{a} - 4 \sqrt{a}$

Solution

$−3 \sqrt{a}$

$\sqrt{b} - 6 \sqrt{b}$

$5 \sqrt{c} + 2 \sqrt{c}$

Solution

$7 \sqrt{c}$

$7 \sqrt{d} + 2 \sqrt{d}$

$8 \sqrt{a} - 2 \sqrt{b}$

Solution

$8 \sqrt{a} - 2 \sqrt{b}$

$5 \sqrt{c} - 3 \sqrt{d}$

$5 \sqrt{m} + \sqrt{n}$

Solution

$5 \sqrt{m} + \sqrt{n}$

$\sqrt{n} + 3 \sqrt{p}$

$8 \sqrt{7} + 2 \sqrt{7} + 3 \sqrt{7}$

Solution

$13 \sqrt{7}$

$6 \sqrt{5} + 3 \sqrt{5} + \sqrt{5}$

$3 \sqrt{11} + 2 \sqrt{11} - 8 \sqrt{11}$

Solution

$−3 \sqrt{11}$

$2 \sqrt{15} + 5 \sqrt{15} - 9 \sqrt{15}$

$3 \sqrt{3} - 8 \sqrt{3} + 7 \sqrt{5}$

Solution

$−5 \sqrt{3} + 7 \sqrt{5}$

$5 \sqrt{7} - 8 \sqrt{7} + 6 \sqrt{3}$

$6 \sqrt{2} + 2 \sqrt{2} - 3 \sqrt{5}$

Solution

$8 \sqrt{2} - 3 \sqrt{5}$

$7 \sqrt{5} + \sqrt{5} - 8 \sqrt{10}$

$3 \sqrt{2 a} - 4 \sqrt{2 a} + 5 \sqrt{2 a}$

Solution

$4 \sqrt{2 a}$

$\sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b}$

$8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c}$

Solution

$\sqrt{3 c}$

$3 \sqrt{5 d} + 8 \sqrt{5 d} - 11 \sqrt{5 d}$

$5 \sqrt{3 a b} + \sqrt{3 a b} - 2 \sqrt{3 a b}$

Solution

$4 \sqrt{3 a b}$

$8 \sqrt{11 c d} + 5 \sqrt{11 c d} - 9 \sqrt{11 c d}$

$2 \sqrt{p q} - 5 \sqrt{p q} + 4 \sqrt{p q}$

Solution

$\sqrt{p q}$

$11 \sqrt{2 r s} - 9 \sqrt{2 r s} + 3 \sqrt{2 r s}$

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{50} + 4 \sqrt{2}$

Solution

$9 \sqrt{2}$

$\sqrt{48} + 2 \sqrt{3}$

$\sqrt{80} - 3 \sqrt{5}$

Solution

$\sqrt{5}$

$\sqrt{28} - 4 \sqrt{7}$

$\sqrt{27} - \sqrt{75}$

Solution

$−2 \sqrt{3}$

$\sqrt{72} - \sqrt{98}$

$\sqrt{48} + \sqrt{27}$

Solution

$7 \sqrt{3}$

$\sqrt{45} + \sqrt{80}$

$2 \sqrt{50} - 3 \sqrt{72}$

Solution

$−8 \sqrt{2}$

$3 \sqrt{98} - \sqrt{128}$

$2 \sqrt{12} + 3 \sqrt{48}$

Solution

$16 \sqrt{3}$

$4 \sqrt{75} + 2 \sqrt{108}$

$\frac{2}{3} \sqrt{72} + \frac{1}{5} \sqrt{50}$

Solution

$5 \sqrt{2}$

$\frac{2}{5} \sqrt{75} + \frac{3}{4} \sqrt{48}$

$\frac{1}{2} \sqrt{20} - \frac{2}{3} \sqrt{45}$

Solution

$− \sqrt{5}$

$\frac{2}{3} \sqrt{54} - \frac{3}{4} \sqrt{96}$

$\frac{1}{6} \sqrt{27} - \frac{3}{8} \sqrt{48}$

Solution

$− \sqrt{3}$

$\frac{1}{8} \sqrt{32} - \frac{1}{10} \sqrt{50}$

$\frac{1}{4} \sqrt{98} - \frac{1}{3} \sqrt{128}$

Solution

$- \frac{11}{12} \sqrt{2}$

$\frac{1}{3} \sqrt{24} + \frac{1}{4} \sqrt{54}$

$\sqrt{72 a^{5}} - \sqrt{50 a^{5}}$

Solution

$a^{2} \sqrt{2 a}$

$\sqrt{48 b^{5}} - \sqrt{75 b^{5}}$

$\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$

Solution

$2 c^{3} \sqrt{5 c}$

$\sqrt{96 d^{9}} - \sqrt{24 d^{9}}$

$9 \sqrt{80 p^{4}} - 6 \sqrt{98 p^{4}}$

Solution

$36 p^{2} \sqrt{5} - 42 p^{2} \sqrt{2}$

$8 \sqrt{72 q^{6}} - 3 \sqrt{75 q^{6}}$

$2 \sqrt{50 r^{8}} + 4 \sqrt{54 r^{8}}$

Solution

$10 r^{4} \sqrt{2} + 12 r^{4} \sqrt{6}$

$5 \sqrt{27 s^{6}} + 2 \sqrt{20 s^{6}}$

$3 \sqrt{20 x^{2}} - 4 \sqrt{45 x^{2}} + 5 x \sqrt{80}$

Solution

$14 x \sqrt{5}$

$2 \sqrt{28 x^{2}} - \sqrt{63 x^{2}} + 6 x \sqrt{7}$

$3 \sqrt{128 y^{2}} + 4 y \sqrt{162} - 8 \sqrt{98 y^{2}}$

Solution

$4 y \sqrt{2}$

$3 \sqrt{75 y^{2}} + 8 y \sqrt{48} - \sqrt{300 y^{2}}$

Mixed Practice

$2 \sqrt{8} + 6 \sqrt{8} - 5 \sqrt{8}$

Solution

$6 \sqrt{2}$

$\frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48}$

$\sqrt{175 k^{4}} - \sqrt{63 k^{4}}$

Solution

$2 k^{2} \sqrt{7}$

$\frac{5}{6} \sqrt{162} + \frac{3}{16} \sqrt{128}$

$2 \sqrt{363} - 2 \sqrt{300}$

Solution

$2 \sqrt{3}$

$\sqrt{150} + 4 \sqrt{6}$

$9 \sqrt{2} - 8 \sqrt{2}$

Solution

$\sqrt{2}$

$5 \sqrt{x} - 8 \sqrt{y}$

$8 \sqrt{13} - 4 \sqrt{13} - 3 \sqrt{13}$

Solution

$\sqrt{13}$

$5 \sqrt{12 c^{4}} - 3 \sqrt{27 c^{6}}$

$\sqrt{80 a^{5}} - \sqrt{45 a^{5}}$

Solution

$a^{2} \sqrt{5 a}$

$\frac{3}{5} \sqrt{75} - \frac{1}{4} \sqrt{48}$

$21 \sqrt{19} - 2 \sqrt{19}$

Solution

$19 \sqrt{19}$

$\sqrt{500} + \sqrt{405}$

$\frac{5}{6} \sqrt{27} + \frac{5}{8} \sqrt{48}$

Solution

$5 \sqrt{3}$

$11 \sqrt{11} - 10 \sqrt{11}$

$\sqrt{75} - \sqrt{108}$

Solution

$− \sqrt{3}$

$2 \sqrt{98} - 4 \sqrt{72}$

$4 \sqrt{24 x^{2}} - \sqrt{54 x^{2}} + 3 x \sqrt{6}$

Solution

$8 x \sqrt{6}$

$8 \sqrt{80 y^{6}} - 6 \sqrt{48 y^{6}}$

Everyday Math

A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression $9 (8 \sqrt{2}) + 8 (2 \sqrt{2})$ . (Round to the nearest tenth of an inch.)

Solution

$124.5 inches$

Suzy wants to use square tiles on the border of a spa she is installing in her backyard. She will use large tiles that have area of 12 square inches, medium tiles that have area of 8 square inches, and small tiles that have area of 4 square inches. Once section of the border will require 4 large tiles, 8 medium tiles, and 10 small tiles to cover the width of the wall. Simplify the expression $4 \sqrt{12} + 8 \sqrt{8} + 10 \sqrt{4}$ to determine the width of the wall. (Round to the nearest tenth of an inch.)

Writing Exercises

Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.

Solution

Answers will vary.

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and three rows. The columns are labeled, “I can…,” “Confidently,” “With some help,” and “No – I don’t get it!” Under the “I can…” column the rows read, “add and subtract like square roots.,” and “add and subtract square roots that need simplification.” The other rows under the other columns are empty.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

like square roots

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