Square Roots & Cube Roots
Master important formulas and concepts with our comprehensive guide
Important Formulas & Concepts
Study MaterialSquare Roots & Cube Roots
Square Roots and Cube Roots are important topics in Quantitative Aptitude and are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and various competitive examinations.
This chapter helps students improve calculation speed, understand number properties, simplify mathematical expressions, and solve arithmetic as well as algebraic problems efficiently.
Why Square Roots & Cube Roots are Important?
- Frequently asked in simplification and arithmetic questions.
- Improves calculation speed in aptitude exams.
- Useful in algebra, geometry, and mensuration.
- Helps solve approximation and simplification problems quickly.
- Important for understanding powers and exponents.
What is a Square Root?
If x² = y, then x is called the square root of y.
The square root of a number is denoted by:
√y = x
| Number | Square Root |
|---|---|
| 4 | 2 |
| 9 | 3 |
| 196 | 14 |
| 625 | 25 |
What is a Cube Root?
The cube root of a number is the number whose cube gives the original number.
If x³ = y, then:
∛y = x
| Number | Cube Root |
|---|---|
| 8 | 2 |
| 27 | 3 |
| 125 | 5 |
| 343 | 7 |
Perfect Squares
A perfect square is a number obtained by squaring an integer.
| Number | Square |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 10 | 100 |
| 15 | 225 |
| 20 | 400 |
✔ A perfect square always has even powers in its prime factorization.
Perfect Cubes
A perfect cube is a number obtained by cubing an integer.
| Number | Cube |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 10 | 1000 |
✔ A perfect cube always has powers in multiples of 3 in its prime factorization.
Properties of Square Roots
| Property | Formula |
|---|---|
| Product Property | √(xy) = √x × √y |
| Division Property | √(x/y) = √x / √y |
| Square Root of Square | √(x²) = x |
| Exponent Form | √x = x1/2 |
Example:
√(25 × 16)
= √25 × √16
= 5 × 4
= 20
Properties of Cube Roots
| Property | Formula |
|---|---|
| Product Property | ∛(xy) = ∛x × ∛y |
| Division Property | ∛(x/y) = ∛x / ∛y |
| Cube Root of Cube | ∛(x³) = x |
| Exponent Form | ∛x = x1/3 |
Example:
∛(8 × 27)
= ∛8 × ∛27
= 2 × 3
= 6
Methods to Find Square Root
1. Prime Factorization Method
Express the number as product of prime factors and make pairs of identical factors.
Example:
Find √144
144 = 2 × 2 × 2 × 2 × 3 × 3
Pair identical factors:
(2 × 2)(2 × 2)(3 × 3)
√144 = 2 × 2 × 3
= 12
2. Long Division Method
Used to find square roots of large numbers and non-perfect squares.
Digits are grouped in pairs from right to left.
Methods to Find Cube Root
Prime Factorization Method
Express the number as product of prime factors and make groups of three identical factors.
Example:
Find ∛1728
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
Group factors:
(2 × 2 × 2)(2 × 2 × 2)(3 × 3 × 3)
∛1728 = 2 × 2 × 3
= 12
Square Root of Decimal Numbers
To find square root of decimal numbers:
- Convert decimal into fraction if needed.
- Apply square root separately to numerator and denominator.
- Place decimal point carefully.
Example:
√0.25
= √(25/100)
= 5/10
= 0.5
Cube Root of Decimal Numbers
Group decimal digits in sets of three while solving cube roots of decimals.
Example:
∛0.008
= ∛(8/1000)
= 2/10
= 0.2
Important Algebraic Identities
| Identity | Formula |
|---|---|
| Square of Sum | (a + b)² = a² + b² + 2ab |
| Square of Difference | (a − b)² = a² + b² − 2ab |
| Difference of Squares | (a + b)(a − b) = a² − b² |
| Cube of Sum | (a + b)³ = a³ + b³ + 3ab(a + b) |
| Cube of Difference | (a − b)³ = a³ − b³ − 3ab(a − b) |
Important Formulae
| Concept | Formula |
|---|---|
| Square Root Product Rule | √(xy) = √x × √y |
| Square Root Division Rule | √(x/y) = √x / √y |
| Cube Root Product Rule | ∛(xy) = ∛x × ∛y |
| Cube Root Division Rule | ∛(x/y) = ∛x / ∛y |
| Exponent Representation | √x = x1/2, ∛x = x1/3 |
Important Exam Tips
- Memorize squares up to 50 and cubes up to 20.
- Practice prime factorization regularly.
- Learn shortcut methods for perfect squares and cubes.
- Be careful while placing decimal points.
- Use algebraic identities for faster calculations.
- Remember square roots involve pairs while cube roots involve triplets.
- Practice approximation techniques for non-perfect squares.
Square Roots and Cube Roots are essential topics in Quantitative Aptitude. Strong understanding of roots, powers, prime factorization, and algebraic identities helps candidates solve aptitude questions quickly and accurately in competitive examinations.