Square Roots and Cube Roots: Easy Methods & Examples

Introduction

Calculating square roots and cube roots can seem challenging, but with the right techniques, you can quickly determine the square or cube root of perfect and non-perfect numbers. In this article, we’ll cover:

Simple tricks to find square roots and cube roots
Step-by-step explanations with examples
Quick mental math techniques
Solved problems for practice

Want to learn an easy way to calculate square and cube roots? Keep reading!

Finding Square Roots: Easy Trick

The square root of a number is a value that, when multiplied by itself, gives the original number.

For example:

√81 = 9 because 9 × 9 = 81
√1369 = 37 because 37 × 37 = 1369
√729 = 27 because 27 × 27 = 729

Example: Finding the Square Root of 2209

Identify the last digit (unit digit) of 2209: It is 9
A perfect square ending in 9 can have square roots ending in 3 or 7 (since 3² = 9 and 7² = 49).
Identify the range: 2209 is between 40² (1600) and 50² (2500).
Check the closer value: 45² = 2025, and 2209 is greater than 2025, so the correct answer is 47.

Answer: √2209 = 47

Finding Cube Roots: Quick Method

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

For example:

∛27 = 3 because 3 × 3 × 3 = 27
∛64 = 4 because 4 × 4 × 4 = 64
∛729 = 9 because 9 × 9 × 9 = 729

Example: Finding the Cube Root of 389017

Identify the last digit of 389017: It is 7.
- From the cube table, we know that only 3³ = 27 ends in 7, so the last digit of the cube root is 3.
Identify the range: 389 lies between 7³ (343) and 8³ (512).
The first digit must be 7 (since 389 is closer to 343).

Answer: ∛389017 = 73

Solved Examples

Example 1: Cube Root of 314432

Last digit is 2, and only 8³ = 512 ends in 2, so the last digit of the cube root is 8.
The first three digits (314) are between 6³ (216) and 7³ (343), so the first digit is 6.

Answer: ∛314432 = 68

Example 2: Square Root of 3844

Last digit is 4, so the square root must end in 2 or 8.
3844 lies between 60² (3600) and 70² (4900).
62² = 3844, so the square root is 62.

Answer: √3844 = 62

Example 3: Smallest Perfect Square Divisible by 21, 36, and 66

Find the LCM of 21, 36, and 66:
LCM = 2 × 2 × 3 × 3 × 7 × 11 = 2772.
Make it a perfect square by multiplying with missing factors:
2772 × 7 × 11 = 213444.

Answer: 213444 is the required perfect square.

Example 4: Solving for Square Root in an Expression

Given: (4a² – 4a + 1)⁰.⁵ + 3a, where a = 0.1039

Rewrite: (1 – 2a)²⁰.⁵ + 3a
Simplify: (1 – 2a) + 3a
Substitute a = 0.1039:

Answer: 1.1039

Example 5: Solving a Cube Root Equation

Given: (0.0169x)⁰.⁵ = 1.3

Square both sides: 0.0169x = 1.69
Solve for x:

Answer: x = 100

Commonly Asked Questions on Square & Cube Roots

1) How many two-digit numbers have a square ending in 8?

Answer: None. A number ending in 8 can never be a perfect square.

2) Cube Root of 0.000216?

Convert to a fraction: (0.000216) = (216/1000000)
Take cube root: ∛216/∛1000000 = 6/100 = 0.06

Answer: 0.06

3) Cube Root of 2197?

Factorize: 2197 = 13 × 13 × 13
Cube root = 13

Answer: 13

4) Smallest Number to Multiply 600 to Make It a Perfect Square?

Factorize: 600 = 2 × 2 × 2 × 3 × 5 × 5
Multiply by 3 × 2 = 6 to balance factors.

Answer: 6

5) Arranging People in a Perfect Square Formation

A person has 2410 people but finds 9 extra when forming a square.

Equation: (x × x) + 9 = 2410
Solving: x = 49

Answer: 49 people per row

Conclusion

Mastering square roots and cube roots can significantly improve your problem-solving speed in competitive exams and daily calculations. Using these tricks, you can:

Quickly identify perfect squares and cubes
Apply mental math techniques for fast calculations
Solve square root and cube root problems efficiently