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Quizzes > Mathematics

Adding and Subtracting Radicals Quiz: Combine Like Square Roots

Quick, free radicals quiz to add and subtract square roots. Instant results.

Editorial: Review CompletedCreated By: Emily RandallUpdated Aug 25, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art quiz illustration with radical symbols adding and subtracting radicals on golden yellow background

This quiz helps you practice adding and subtracting radicals, spot when terms are like radicals, and simplify square roots with confidence. Work through short problems with instant feedback, then explore square root practice, try the simplifying nonperfect roots quiz, or build fluency with simplifying expressions practice.

What is ?5 + ?5?
5?2
?10
2?5
10
Since ?5 and ?5 are like terms, you can add their coefficients: 1?5 + 1?5 = 2?5. You do not multiply the radicands when adding like radicals. See more at .
Simplify 3?3 + 2?3.
6?3
5?3
?6
?5
Both terms share the radical ?3, so you add their coefficients: 3 + 2 = 5, giving 5?3. Adding unlike radicands would not be valid. See .
What is ?12 + ?3 in simplest form?
2?6
3?3
?9
?15
First simplify ?12 to 2?3. Then 2?3 + 1?3 = 3?3. You can only add like radicals after simplifying. For more detail, see .
Simplify 5?2 - 2?2.
?2
7?2
?10
3?2
Subtract the coefficients of like radicals: 5 - 2 = 3, so the result is 3?2. You cannot subtract the radicands themselves. See .
What is ?18 - ?8 simplified?
?10
2?6
?2
?16
Simplify each: ?18 = 3?2 and ?8 = 2?2. Then subtract: 3?2 - 2?2 = 1?2 = ?2. Learn more at .
Compute ??7 + 4?7.
?3?7
?7
3?7
5?7
Combine like terms: (?1 + 4)?7 = 3?7. You handle negative coefficients by standard subtraction. See details at .
Simplify ?50 + ?18.
?68
7?2
5?5
8?2
Write ?50 as 5?2 and ?18 as 3?2. Then add coefficients: 5 + 3 = 8, giving 8?2. See .
What is 2?6 + 3?2?
5?3
2?6 + 3?2
5?8
?12 + ?18
?6 and ?2 are unlike radicals, so they cannot be combined. The expression remains 2?6 + 3?2. For more, visit .
Simplify ?45 + ?20.
3?4 + 2?5
5?5
?65
?25 + ?4
Rewrite ?45 as 3?5 and ?20 as 2?5, then add to get 5?5. Only like radicals may be combined. See .
What is ?72 ? ?8 in simplest form?
4?2
6?10
2?3
?64
Simplify ?72 to 6?2 and ?8 to 2?2, then subtract: 6?2 ? 2?2 = 4?2. See .
Simplify 2?75 + ?27.
13?3
10?5
?2025
5?9
2?75 = 2·5?3 = 10?3; ?27 = 3?3. Add to get 13?3. You must simplify each radical first. More at .
What is 3?12 ? 2?27?
0
?3
?9
?324
3?12 = 3·2?3 = 6?3; 2?27 = 2·3?3 = 6?3; subtract yields 0. Always simplify before combining. See .
Simplify ?32 + ?18 ? ?2.
6?2
?48
?72
5?3
?32 = 4?2, ?18 = 3?2. So 4?2 + 3?2 ? 1?2 = 6?2. Simplify each radicand first. More info at .
What is 5?8 + 3?2 simplified?
?40
2?5
8?10
13?2
5?8 = 5·2?2 = 10?2. Add 3?2 to get 13?2. Like radicals must share the same simplified form. See .
Simplify ?75 ? 2?3.
3?3
5?5
?72
?225
?75 = 5?3. Then 5?3 ? 2?3 = 3?3. You simplify radicands before combining. More at .
What is ?98 + ?32 ? ?18 simplified?
2?7
8?2
7?9
?148
?98 = 7?2, ?32 = 4?2, ?18 = 3?2. Sum: 7?2 + 4?2 ? 3?2 = 8?2. Always convert before adding. See .
Simplify 4?50 ? 3?18 + 2?8.
?(2·50)
8?2
15?2
?320
4?50 = 4·5?2 = 20?2; 3?18 = 3·3?2 = 9?2; 2?8 = 2·2?2 = 4?2. So 20?2 ? 9?2 + 4?2 = 15?2. See .
What is ?27 + ?75 ? 2?12 simplified?
4?3
8?3
5?5
?(27+75?48)
?27=3?3, ?75=5?3, 2?12=2·2?3=4?3. So 3?3+5?3?4?3=4?3. Only like terms combine. More at .
Simplify 6?3 + 4?12 ? 5?27.
?432
??3
5?3
15?3
6?3 + 4·2?3 ? 5·3?3 = 6?3 + 8?3 ? 15?3 = ?1?3 = ??3. Combine like radicals carefully. See .
What is 2?50 ? 4?8 + ?18 simplified?
2?5
6?2
5?2
?(100?64+18)
2?50=2·5?2=10?2; 4?8=4·2?2=8?2; ?18=3?2. So 10?2 ? 8?2 + 3?2 = 5?2. See .
Simplify ?162 ? ?32.
7?2
?(162?32)
4?3
5?2
?162=9?2; ?32=4?2. Then 9?2 ? 4?2 = 5?2. Simplify radicands before subtraction. See .
What is 3?20 ? ?45 + 2?5 simplified?
?(180?45+20)
4?5
6?2
5?5
3?20=3·2?5=6?5; ?45=3?5; 2?5=2?5. So 6?5 ? 3?5 + 2?5 = 5?5. Combine like radicals only. See .
Simplify ?50 + ?8 ? ?18 + ?2.
5?2
4?2
?(50+8?18+2)
6?2
Convert: ?50=5?2, ?8=2?2, ?18=3?2, ?2=1?2. Sum: 5 + 2 ? 3 + 1 = 5, giving 5?2. For a deep dive, see .
What is 2?3 + ?12 ? ?27 + 3?75 simplified?
12?3
10?3
14?3
16?3
2?3 + 2?3 ? 3?3 + 3·5?3 = (2+2?3+15)?3 = 16?3. Always simplify and then combine coefficients of like radicals. See .
Simplify ?72 + ?18 ? ?8 ? ?2 + ?32.
8?2
9?2
11?2
10?2
?72=6?2, ?18=3?2, ?8=2?2, ?2=1?2, ?32=4?2. Then 6 + 3 ? 2 ? 1 + 4 = 10, so 10?2. Learn more at .
0
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Study Outcomes

  1. Identify Like Radicals -

    Recognize which radical expressions share the same index and radicand so you can determine when adding radicals is possible.

  2. Simplify Radical Terms -

    Apply rules to simplify radicals into their simplest form before performing any operations on them.

  3. Add Radical Expressions -

    Use proper techniques to combine like terms and add radicals accurately in a variety of expressions.

  4. Subtract Radical Expressions -

    Execute subtraction of like radical terms, handling signs and coefficients to maintain correct simplification.

  5. Combine and Simplify -

    Integrate adding radicals and subtracting radicals in multi-step problems and simplify radicals fully.

  6. Apply Instant Feedback -

    Leverage quiz feedback and explanations to correct mistakes and reinforce your mastery of radical expressions.

Cheat Sheet

  1. Like Radicals Rule -

    Only radicals with the same index and exactly the same radicand can be added or subtracted. For example, √8+3√8=4√8, but √8+√18 cannot combine directly because 8≠18 (source: Khan Academy).

  2. Simplify Before Combining -

    Always factor out perfect squares (or cubes for higher indices) to simplify each radical before adding or subtracting. For instance, √18=3√2 and √8=2√2, making it easy to see 3√2+2√2=5√2 (source: MIT OpenCourseWare).

  3. Adding Radicals -

    Treat like radicals as you would like terms in algebra: combine coefficients and keep the radical part intact. Example: 2√3+5√3=7√3, just like 2x+5x=7x (source: Stewart's Calculus).

  4. Subtracting Radicals -

    Subtraction follows the same procedure as addition: ensure radicals are like, then subtract their numerical coefficients. For example, 6√5−2√5=4√5 once you confirm both terms share √5 (source: University of Arizona Math).

  5. Think "√a as x" Mnemonic -

    Visualize each radical √a as a single variable x to reinforce combining rules: x+2x=3x or x−x=0. This trick makes it easier to remember "can you add to radicals?" by treating them like algebraic terms (source: Paul's Online Math Notes).

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