Square Root and Cube Root Notes for SSC Exams

Square Root and Cube Root questions appear in the simplification and approximation sets of SSC Tier 1 and Tier 2. If you know the squares and cubes by heart, you can solve these questions in seconds. These square root and cube root notes for SSC explain the methods, the unit digit rules and the squaring tricks with solved examples.

Squares and Cubes to Remember

Also remember the squares from 17 to 30: 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841 and 900.

Unit Digit Rules

A perfect square can only end in 0, 1, 4, 5, 6 or 9. A number ending in 2, 3, 7 or 8 is never a perfect square. For a cube root, the last digit of the cube decides the last digit of the root directly.

Methods to Find Roots

• Prime factor method: break the number into prime factors. For a square root, take one from each pair. For a cube root, take one from each group of three.
• Unit digit and estimation: for a perfect square like 1764, 40 squared is 1600 and 50 squared is 2500, so the root is between 40 and 50. The last digit 4 means the root ends in 2 or 8, so check 42.
• For decimals: count the decimal places. The square root has half the decimal places, and the cube root has one-third.

Squaring Tricks

• Numbers near 100: 99 squared = (100 - 1) squared = 10000 - 200 + 1 = 9801.
• Numbers ending in 5: for (n5) squared, multiply n by (n + 1) and write 25 after it. 35 squared = 3 x 4 = 12, then 25, so 1225.
• Numbers near 50: use (50 + a) squared = 2500 + 100a + a squared.

Solved Examples

Example 1: Find the square root of 1,764.
Solution: 1764 is between 40 squared (1600) and 50 squared (2500). The last digit is 4, so the root ends in 2 or 8. 42 squared = 1764. So the root is 42.

Example 2: Find the cube root of 5,832.
Solution: The last digit 2 means the root ends in 8. 5832 lies between 17 cubed (4913) and 19 cubed (6859), so the tens digit is 1, and 18 cubed = 5832. So the root is 18.

Example 3: Find the square root of 0.0144.
Solution: 0.0144 = 144 / 10000. Root = 12 / 100 = 0.12.

Example 4: Find the value of 99 squared.
Solution: 99 squared = (100 - 1) squared = 10000 - 200 + 1 = 9,801.

Example 5: Find the smallest number by which 180 must be multiplied to get a perfect square.
Solution: 180 = 2 squared x 3 squared x 5. The factor 5 has no pair, so multiply by 5.

Example 6: Find the cube root of 0.027.
Solution: 0.027 = 27 / 1000. Cube root = 3 / 10 = 0.3.

Quick Tips to Solve Faster

• Learn the squares up to 30 and the cubes up to 15 by heart.
• Use the last digit to cut down the options for a root.
• For numbers ending in 5, use the n(n + 1) rule followed by 25.
• For decimals, convert into a fraction and then find the root of the numerator and denominator.
• To make a number a perfect square or cube, look at the unpaired prime factors.

Common Mistakes to Avoid

• Forgetting to halve the decimal places for the square root.
• Taking one factor from each group of two for a cube root.
• Choosing a root whose last digit does not match the unit digit rule.
• Mixing the square and cube values of the numbers 12, 13 and 14.
• Not checking the answer by squaring or cubing it.

Quick Revision Points

• Squares up to 30 and cubes up to 15 should be known.
• Perfect squares end only in 0, 1, 4, 5, 6 and 9.
• Square root: pairs of prime factors. Cube root: triplets of prime factors.
• (n5) squared = n(n + 1) followed by 25.
• Decimal places: halve for square roots and divide by three for cube roots.

Practice Questions With Answers

1. Find the square root of 2,025.
   Answer: 45 (45 x 45).
2. Find the cube root of 2,744.
   Answer: 14 (14 cubed).
3. Find the value of 105 squared.
   Answer: 11,025 ((100 + 5) squared).
4. Find the square root of 1.69.
   Answer: 1.3 (169 / 100).
5. Find the smallest number by which 72 must be multiplied to get a perfect cube.
   Answer: 3 (72 = 2 cubed x 3 squared).
6. Find the smallest number by which 200 must be divided to get a perfect square.
   Answer: 2 (200 = 2 cubed x 5 squared).
7. Find the square root of 0.0049.
   Answer: 0.07 (49 / 10000).
8. Find the value of 48 squared.
   Answer: 2,304 ((50 - 2) squared).
9. Find the cube root of 125 / 343.
   Answer: 5 / 7 (5 cubed / 7 cubed).
10. What is the nearest whole number to the square root of 50?
    Answer: 7 (7 squared = 49).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC simplification practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Simplification and BODMAS Notes
• Number System Notes
• Algebra Formulas Notes
• Surds and Indices Notes
• SSC Simplification Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is a perfect square?

A perfect square is a number that is the square of a whole number, such as 16, 25 and 144.

What is a perfect cube?

A perfect cube is a number that is the cube of a whole number, such as 8, 27 and 125.

How do I find the square root quickly?

Use the unit digit rule and the range of squares of tens (for example 40 squared and 50 squared) to find the root.

How do I find the cube root quickly?

The last digit of the cube tells the last digit of the root, and the thousands part tells the tens digit.

Which numbers can never be perfect squares?

Numbers ending in 2, 3, 7 or 8 are never perfect squares.

How do I square a number ending in 5?

Multiply the digits before 5 by the next number and write 25 at the end, for example 35 squared = 1225.

How many questions come from roots in SSC exams?

Several questions in simplification and approximation sets use square and cube roots.

How should I practise roots?

Memorise the table, practise the tricks in these notes and attempt the simplification practice paper.