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Quizzes > Mathematics > Basic Math & Arithmetic

Square Roots Quiz: Perfect vs Imperfect Squares - Challenge Yourself!

Think you can ace this imperfect square root quiz? Dive in now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
paper art numbers and square root symbols on dark blue background for quiz on perfect and imperfect square roots

This quiz helps you master square root imperfect squares by estimating roots of non-perfect numbers and telling them from perfect squares. Warm up with a quick review of squares and roots, then try the full square root quiz to build speed and spot gaps before a test.

Which of the following integers is a perfect square?
15
36
50
44
A perfect square is an integer that can be expressed as the product of an integer multiplied by itself. 36 is 6 times 6, so it is a perfect square. The numbers 15, 44, and 50 do not have integer square roots and therefore are not perfect squares. For more information, visit .
Which of these numbers has an integer square root?
80
81
82
83
An integer square root occurs when the square root of a number is a whole number. 81 equals 9 times 9, giving an integer square root of 9. The numbers 80, 82, and 83 do not yield integer square roots. See for more details.
Which of the following numbers is not a perfect square?
144
100
150
121
100, 121, and 144 are perfect squares (10², 11², and 12² respectively). 150 cannot be expressed as an integer squared, so it is not a perfect square. Recognizing perfect squares helps in identifying integer roots. More examples at .
What is the simplified form of ?12?
?12
2?3
4?3
3?2
To simplify ?12, factor 12 into 4×3. The square root of 4 is 2, so ?12 becomes 2?3. This process breaks down the radicand into a perfect square times another factor. Learn more about radical simplification at .
Which square root is irrational?
?49
?81
?97
?64
A square root of a non-perfect square is irrational, meaning it cannot be expressed as a fraction. 97 is not a perfect square, so ?97 is irrational. In contrast, ?49, ?64, and ?81 are all integers (7, 8, and 9). See for a deeper dive.
What is the approximate value of ?50 to two decimal places?
7.10
7.07
6.93
7.14
?50 is approximately 7.071067..., which rounds to 7.07 when taken to two decimal places. Accurate rounding of irrational square roots is essential in many calculations. For step-by-step methods, check .
Simplify ?75.
3?5
5?5
7?5
5?3
75 can be factored into 25×3, and the square root of 25 is 5. Thus, ?75 simplifies to 5?3. Identifying the largest perfect square factor is the key step. More on this process at .
What is the integer part (floor) of ?200?
13
14
16
15
?200 is about 14.1421, so the integer part (floor) is 14. Recognizing that ?196 is exactly 14 helps to bound the value. This technique is useful for estimating roots. For more examples, see .
Rationalize the denominator and simplify: 1/?3.
3?3
?3/3
?3
?3/2
To rationalize, multiply numerator and denominator by ?3 to get ?3/(?3×?3) = ?3/3. This removes the radical from the denominator. Rationalizing denominators ensures standard form in algebra. Learn how at .
Simplify ?50 + ?18 - ?8.
6?2
2?2
8?2
4?2
Convert each term: ?50=5?2, ?18=3?2, and ?8=2?2. Adding and subtracting gives (5+3?2)?2 = 6?2. Combining like radical terms simplifies the expression neatly. For more practice, visit .
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Study Outcomes

  1. Identify Perfect and Non-Perfect Squares -

    Quickly distinguish perfect squares from imperfect squares within the context of square root imperfect squares to strengthen your foundational number sense.

  2. Calculate Exact Square Roots -

    Compute precise roots for perfect squares using the perfect squares quiz methodology to build confidence in exact calculations.

  3. Estimate Roots of Nonperfect Squares -

    Apply estimation strategies to approximate square roots of nonperfect squares and improve your accuracy with the square root of nonperfect squares.

  4. Evaluate Quiz Feedback -

    Use instant feedback from the imperfect square root quiz to assess your solutions, correct errors, and refine your problem-solving approach.

  5. Enhance Computational Fluency -

    Practice repeatedly with non-perfect squares practice to boost speed and precision in square root calculations.

  6. Strengthen Exam Readiness -

    Solidify your skills and confidence for upcoming math assessments by mastering challenging square root problems.

Cheat Sheet

  1. Identifying Perfect vs Imperfect Squares -

    To master square root imperfect squares, begin by memorizing perfect squares up to 15² (1, 4, 9, …, 225). Use the "Last-Digit" trick: perfect squares end in 0, 1, 4, 5, 6, or 9, so any number ending in 2, 3, 7, or 8 is instantly non-perfect. This quick check boosts your accuracy in both perfect squares quiz and non-perfect squares practice.

  2. Prime Factorization Simplification -

    Breaking down the radicand via prime factorization makes simplifying square roots of nonperfect squares a breeze. For example, √72 = √(2²·3²·2) = 2·3·√2 = 6√2. This method, backed by university-level algebra texts, ensures you effortlessly extract all perfect factors.

  3. Decimal Approximation with Newton's Method -

    Newton-Raphson iteration is a powerful tool for calculating the square root of nonperfect squares to any desired precision. Start with a guess x₀ and apply xₙ₊₝ = (xₙ + S/xₙ)/2; for S=20 and x₀=4.5, you'll quickly hone in on √20≈4.4721. Many academic sources recommend this for high-speed, accurate approximations in an imperfect square root quiz.

  4. Irrationality of Non-Square Roots -

    One of the cornerstones of number theory is that the square root of any nonperfect square integer is irrational (no exact fraction representation). This theorem dates back to Euclid's Elements and underscores why √2, √3, and √5 have non-repeating, non-terminating decimals. Embracing this concept strengthens your foundation for advanced mathematics.

  5. Estimating Between Consecutive Squares -

    When faced with an imperfect square root, locate the two nearest perfect squares to bracket your value. For instance, √50 lies between 7 (7²=49) and 8 (8²=64), so √50≈7.07. This "between squares" strategy is a fast mnemonic and key aspect of any perfect squares quiz or non-perfect squares practice session.

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