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Quizzes > Mathematics > Basic Math & Arithmetic

Squares & Square Roots Quiz: Test Your Skills

Think you can ace these square and square root questions? Dive in!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration with numbers and square root symbols on dark blue background for quiz on squares and square roots

Use this squares and square roots quiz to practice squaring numbers and finding roots, and spot any gaps before a test. You get instant feedback on each item. For more, try the quick square root quiz or see how this connects to the Pythagorean theorem .

What is 7 squared?
56
49
14
36
Squaring a number means multiplying it by itself. Thus, 7 squared is 7 × 7 = 49. This operation corresponds to the area of a square with side length 7. .
What is the square root of 81?
7
81
9
8
The square root of a number is the value that, when multiplied by itself, gives the original number. Since 9 × 9 equals 81, the square root is 9. We refer to the principal square root, which is non-negative. .
What is (-3) squared?
9
6
-9
-6
Squaring a negative number multiplies the number by itself, and a negative times a negative yields a positive. Thus, (-3) × (-3) = 9. We always square the full value including the negative sign. .
Which of the following is a perfect square?
49
52
55
50
A perfect square is an integer that is the square of another integer. Since 7 × 7 = 49, it is a perfect square. The other options are not the product of any integer multiplied by itself. .
What is the square root of 121?
121
11
10
12
The square root of 121 is the number which when squared gives 121. Since 11 × 11 = 121, the square root is 11. We consider the principal square root, which is non-negative. .
What is the square of 5?
10
15
20
25
Squaring a number means multiplying it by itself. Thus, 5 × 5 equals 25. This represents the area of a square with side length 5. .
What is the square root of 144?
144
12
14
16
Since 12 × 12 = 144, the principal square root of 144 is 12. Square roots are non-negative values by convention. .
What is the square root of 169?
169
12
13
14
The principal square root of 169 is the non-negative number whose square equals 169. Since 13 × 13 = 169, the square root is 13. .
What is (2.5) squared?
5
6.25
6
25
To square a decimal, multiply the number by itself: 2.5 × 2.5 = 6.25. Decimals follow the same squaring rule as whole numbers, with decimal placement in the product determined by the sum of places in each factor. .
What is the square root of 0.49?
7
0.07
0.49
0.7
The square root of a decimal is found by determining which decimal squared equals the original number. Since 0.7 × 0.7 = 0.49, the square root is 0.7. .
What is the approximate value of ?2 to two decimal places?
1.21
1.42
1.44
1.41
?2 is an irrational number approximately equal to 1.4142. Rounded to two decimal places, it is 1.41. This approximation is widely used in geometry and engineering. .
Simplify ?72.
8?3
6?2
3?8
2?72
72 can be factored into 36 × 2, and ?36 = 6. Thus ?72 = ?(36×2) = 6?2. Simplifying radicals involves extracting perfect square factors from under the radical. .
What is the value of (?3)²?
9
3
?3
1
Squaring a square root reverses the radical: (?3)² = 3. This holds true for any non-negative number under a principal square root. .
What is the square root of 1/4?
2
1
1/4
1/2
The square root of a fraction is the square root of numerator over the square root of denominator: ?(1/4) = ?1/?4 = 1/2. By convention, we take the principal (positive) root. .
Which of these numbers is NOT a perfect square?
200
144
169
121
121 = 11², 144 = 12², and 169 = 13², so these are perfect squares. 200 is not the square of any integer (14² = 196, 15² = 225). .
What is the value of (3?2)²?
12
9?2
18
6?2
Square the coefficient and the radical separately: (3)² = 9 and (?2)² = 2, so (3?2)² = 9 × 2 = 18. .
Simplify ?18 + ?8.
5?2
7?2
?10
?26
Break down each radical: ?18 = 3?2 and ?8 = 2?2, so their sum is 3?2 + 2?2 = 5?2. Combining like radical terms simplifies the expression. .
What is 5/?3 when rationalized?
5?3
3?5
?3/5
(5?3)/3
To rationalize, multiply numerator and denominator by ?3: (5×?3)/(?3×?3) = 5?3/3. This removes the radical from the denominator. .
What are the solutions to x² = 16?
4 and -4
-4 only
0
4 only
If x² = 16, then x = ±?16, so x = 4 or x = -4. Equations of the form x² = a have two real solutions when a > 0. .
What is (?5 + ?2)(?5 - ?2)?
3
?10
?3
7
This is a difference of squares: (a+b)(a?b) = a² ? b². Here, a=?5 and b=?2, so a² ? b² = 5 ? 2 = 3. .
What is (2+3)² ? 2² ? 3²?
12
0
-4
25
Compute directly: (5)² = 25, then subtract 2² = 4 and 3² = 9: 25 ? 4 ? 9 = 12. This shows how cross terms contribute in binomial expansion. .
If ?x = 7, what is x?
-7
49
7
14
Squaring both sides gives x = 7² = 49. Since square roots yield the principal (non-negative) root, x must be positive. .
For x ? 0, simplify ?(50x²).
5x?2
5x²?2
10x?2
x?50
Factor inside the radical: 50x² = 25×2×x². Then ?(25) = 5 and ?(x²) = x (for x ? 0), giving 5x?2. .
Simplify ?(5 + 2?6).
?5 + ?6
2?6
?3 + ?2
?6 + 1
Assume ?(5+2?6) can be written as ?a + ?b. Squaring gives a+b+2?(ab)=5+2?6. Matching terms: a+b=5 and ab=6, solved by a=3, b=2, so the expression is ?3+?2. .
Simplify ?3 + ?12 + ?27 + ?48.
5?3
7?3
10?3
9?3
Simplify each term: ?3 remains ?3, ?12=2?3, ?27=3?3, ?48=4?3. Summing coefficients: 1+2+3+4=10, giving 10?3. .
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Study Outcomes

  1. Understand Perfect Squares -

    Develop a clear understanding of what perfect squares are and recognize them within number sets.

  2. Calculate Square Roots Accurately -

    Practice extracting square roots of perfect squares and learn methods to approximate roots for non-perfect squares.

  3. Identify Square and Square Root Relationships -

    Analyze the connection between squares and their corresponding roots to deepen your number sense.

  4. Solve Square and Square Root Questions -

    Apply strategies to tackle a variety of square and square root questions quickly and efficiently.

  5. Master Problem-Solving Techniques -

    Use shortcuts and mental math tricks to accelerate your calculations in the squares and square roots practice quiz.

  6. Evaluate Your Quiz Performance -

    Review your results to identify strengths and areas for improvement in fundamental square roots skills.

Cheat Sheet

  1. Understanding Perfect Squares -

    Perfect squares result when an integer is multiplied by itself, producing values like 1, 4, 9, 16 and up to 225 (15²). Noting that the gaps between squares follow the odd-number sequence (3, 5, 7…) makes it easier to recall and predict new squares during squares and square roots practice. This pattern is a staple in academic courses and underlies many square and square root questions.

  2. Defining Square Roots -

    The square root symbol (√) denotes the principal root, so √49=7, even though algebraically ±7 are solutions to x²=49. Remembering that √a²=|a| helps avoid sign errors in both practice square roots problems and formal equations. This convention is emphasized in high school curricula and supported by collegiate math standards.

  3. Prime Factorization Method -

    Breaking a number into prime factors (e.g., 144=2²×3²) lets you extract square roots by pairing primes: √144=2×3=6. This reliable technique appears in standardized tests and square roots quizzes, offering a systematic alternative to memorization. Universities and math journals endorse it for precise root calculations on non-perfect squares too.

  4. Estimation and Bounding Techniques -

    When values aren't perfect squares, sandwich your root between two known squares: for √50, use √49<√50<√64 so 7<√50<8. Combining this with linear interpolation yields quick approximations, perfect for timed challenges in a squares fundamentals quiz. Many math contests and educational platforms recommend this strategy for rapid mental math.

  5. Using Difference of Squares -

    The identity a² - b²=(a - b)(a+b) can simplify tricky expressions like √(x² - 9) or accelerate mental calculation (e.g., 102² - 98²= (102 - 98)(102+98)=4×200=800). This trick also appears in advanced square and square root questions and problem-solving guides from reputable institutions. Keeping it in your toolkit sharpens both speed and insight for practice square roots problems.

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