SSC Square Root and Cube Root Questions with Answers and Tricks

Square roots and cube roots are used in simplification, mensuration, algebra and number system questions of SSC exams. If you know the squares up to 30 and cubes up to 15, most questions can be solved in a few seconds. This practice paper gives you 21 questions with answers and short solutions.

Practice Paper at a Glance

Square Root and Cube Root in Simple Words

The square root of a number is the value that gives the number when multiplied by itself, and the cube root is the value that gives the number when multiplied three times.

• The square root of 144 is 12 because 12 x 12 = 144.
• The cube root of 343 is 7 because 7 x 7 x 7 = 343.
• A number whose square root is a whole number is a perfect square, and one whose cube root is a whole number is a perfect cube.

Tricks and Shortcuts

• Prime factorisation: pair the factors for a square root and group them in threes for a cube root.
• Unit digit of a perfect square can only be 0, 1, 4, 5, 6 or 9. It can never be 2, 3, 7 or 8.
• Cube roots of 2-digit cubes: the last digit of the cube tells the last digit of the root (1 to 1, 8 to 2, 7 to 3, 4 to 4, 5 to 5, 6 to 6, 3 to 7, 2 to 8, 9 to 9).
• Decimals: the square root of a decimal has half the decimal places of the number, and the cube root has one third.
• Nested roots: if x = square root of (a + x), then x^2 - x - a = 0.
• Square root of a + 2 root b: find p and q with p + q = a and pq = b, the answer is root p + root q.

Set 1: Basic Questions (Q1–Q7)

Q1. Find the square root of 169.
Answer: 13
Solution: 13 x 13 = 169.

Q2. Find the square root of 1,764.
Answer: 42
Solution: 42 x 42 = 1,764.

Q3. Find the cube root of 2,744.
Answer: 14
Solution: 14 x 14 x 14 = 2,744.

Q4. Find the cube root of -125.
Answer: -5
Solution: (-5)^3 = -125.

Q5. Find the square root of 0.0049.
Answer: 0.07
Solution: 0.07 x 0.07 = 0.0049.

Q6. Find the cube root of 0.027.
Answer: 0.3
Solution: 0.3 x 0.3 x 0.3 = 0.027.

Q7. Find the square root of 2,025.
Answer: 45
Solution: 45 x 45 = 2,025.

Set 2: Factorisation and Perfect Squares and Cubes (Q8–Q14)

Q8. By what smallest number should 72 be multiplied to get a perfect square? Also find the square root.
Answer: 2; the square root of 144 is 12
Trick: 72 = 2^3 x 3^2. Multiply by 2 to pair all factors.

Q9. By what smallest number should 108 be divided to get a perfect cube?
Answer: 4
Solution: 108 = 2^2 x 3^3. Dividing by 4 gives 27, whose cube root is 3.

Q10. What is the smallest number to be added to 1,000 to make it a perfect square?
Answer: 24
Solution: 31^2 = 961 and 32^2 = 1,024. So 1,024 - 1,000 = 24.

Q11. What is the smallest number to be subtracted from 1,000 to make it a perfect square?
Answer: 39
Solution: 1,000 - 961 = 39.

Q12. Find the square root of 2.25.
Answer: 1.5
Solution: 1.5 x 1.5 = 2.25.

Q13. Find the cube root of 64/125.
Answer: 4/5
Solution: The cube root of 64 is 4 and of 125 is 5.

Q14. Which digits can never be the unit digit of a perfect square?
Answer: 2, 3, 7 and 8
Trick: No square ends in these digits.

Set 3: Exam-Level Questions (Q15–Q21)

Q15. Find the value of the square root of (6 + square root of (6 + square root of (6 + ...))).
Answer: 3
Trick: x^2 = 6 + x, so x^2 - x - 6 = 0 and x = 3.

Q16. Find the value of the square root of (0.09 x 16).
Answer: 1.2
Solution: 0.3 x 4 = 1.2.

Q17. If the square root of x = 12, find x.
Answer: 144
Solution: x = 12^2 = 144.

Q18. Find the value of the square root of (3 + 2 root 2).
Answer: 1 + root 2
Trick: (1 + root 2)^2 = 3 + 2 root 2.

Q19. Find the cube root of 17,576.
Answer: 26
Solution: 26^3 = 17,576.

Q20. How many perfect squares are there from 100 to 400 (both included)?
Answer: 11
Solution: 10^2 = 100 to 20^2 = 400, so 11 squares.

Q21. Find the value of the square root of (16 x 81 x 25).
Answer: 180
Solution: 4 x 9 x 5 = 180.

Quick Tip for Finding Roots Fast

To find the square root of a perfect square with four digits, look at the last digit to guess the unit digit of the root (for example, a number ending in 4 has a root ending in 2 or 8), and look at the leading digits to find the tens digit. This method gives the answer in about 10 seconds without long division.

Common Mistakes in Square and Cube Roots

• Not memorising the squares and cubes, which makes every question slow.
• Pairing the prime factors wrongly in factorisation.
• Making errors in the decimal places of the root.
• Forgetting that the cube root of a negative number is negative.
• Taking the square root of each term separately in a sum (root of a + b is not root a + root b).
• Rounding the root of a non-perfect square incorrectly.

How to Practice These 21 Questions

1. Memorise squares up to 30 and cubes up to 15.
2. Practise 10 factorisation questions daily.
3. Solve each set in 7 to 8 minutes.
4. Learn the unit digit tricks.
5. Revise the mistakes after one day.

Related Topics to Study Next

• Surds and Indices Questions
• SSC Simplification Questions
• Fractions and Decimals Questions
• SSC Algebra Questions

Frequently Asked Questions

What is a square root?

A square root of a number is a value that gives the number when multiplied by itself.

What is a cube root?

A cube root is a value that gives the number when multiplied by itself three times.

How do I find a square root by factorisation?

Write the prime factors, make pairs of equal factors and take one factor from each pair.

How do I find a cube root by factorisation?

Make groups of three equal factors and take one factor from each group.

What are the possible unit digits of a perfect square?

0, 1, 4, 5, 6 and 9.

How can I find the cube root of a 4 or 5-digit cube quickly?

Use the last digit to find the unit digit of the root and the leading digits to find the tens digit.

Which squares and cubes should I memorise?

Squares up to 30 and cubes up to 15 are enough for SSC exams.

What is the square root of a decimal?

Convert the decimal to a fraction or move the decimal point in pairs. The root has half the decimal places.

How do nested root questions work?

Let the whole expression be x, form an equation like x^2 = a + x and solve it.

Is the cube root of a negative number possible?

Yes. The cube root of -27 is -3.

How many square root questions come in SSC exams?

Usually one or two questions come in simplification and related topics.

How should I prepare roots for SSC exams?

Memorise squares and cubes, learn the tricks and solve simplification and number system questions regularly.