Square Roots & Cube Roots
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Solved Examples
Study MaterialSolved Examples – Square Roots & Cube Roots
Solved examples help students understand the practical application of Square Roots and Cube Roots in competitive examinations. These examples are designed from basic to advanced level and cover important concepts frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement aptitude tests.
Topics Covered in Solved Examples
- Square Root Basics
- Cube Root Basics
- Prime Factorization Method
- Perfect Squares and Perfect Cubes
- Decimal Root Problems
- Algebraic Identities
- Approximation Techniques
- Root Simplification
- Unit Digit Tricks
- Mental Calculation Methods
Example 1: Basic Square Root
Question: Find √196.
Solution:
14 × 14 = 196
Therefore:
√196 = 14
Example 2: Basic Cube Root
Question: Find ∛343.
Solution:
7 × 7 × 7 = 343
Therefore:
∛343 = 7
Example 3: Square Root by Prime Factorization
Question: Find √144 using prime factorization.
Solution:
144 = 2 × 2 × 2 × 2 × 3 × 3
Make pairs:
(2 × 2)(2 × 2)(3 × 3)
Take one number from each pair:
2 × 2 × 3 = 12
Example 4: Cube Root by Prime Factorization
Question: Find ∛1728 using prime factorization.
Solution:
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
Make groups of three:
(2 × 2 × 2)(2 × 2 × 2)(3 × 3 × 3)
Take one number from each group:
2 × 2 × 3 = 12
Example 5: Product Property of Square Root
Question: Simplify:
√(25 × 16)
Solution:
Using:
√(xy) = √x × √y
= √25 × √16
= 5 × 4
= 20
Example 6: Product Property of Cube Root
Question: Simplify:
∛(8 × 27)
Solution:
Using:
∛(xy) = ∛x × ∛y
= ∛8 × ∛27
= 2 × 3
= 6
Example 7: Decimal Square Root
Question: Find √0.25.
Solution:
0.25 = 25/100
√0.25 = √25 / √100
= 5/10
= 0.5
Example 8: Decimal Cube Root
Question: Find ∛0.008.
Solution:
0.008 = 8/1000
∛0.008 = ∛8 / ∛1000
= 2/10
= 0.2
Example 9: Simplifying Surds
Question: Simplify:
√72
Solution:
72 = 36 × 2
√72 = √36 × √2
= 6√2
Therefore:
√72 = 6√2
Example 10: Unit Digit Trick for Square Root
Question: Find √729.
Solution:
729 ends with 9.
Possible unit digit of root = 3 or 7.
Since:
25² = 625
30² = 900
729 lies between them.
Therefore:
√729 = 27
Example 11: Unit Digit Trick for Cube Root
Question: Find ∛2744.
Solution:
Last digit is 4.
Cube root must end with 4.
Ignoring last three digits:
2 lies between 1³ and 2³.
Therefore:
∛2744 = 14
Example 12: Approximation of Square Root
Question: Approximate √50.
Solution:
49 < 50 < 64
√49 = 7
√64 = 8
Therefore:
√50 ≈ 7.07
Example 13: Algebraic Identity
Question: Simplify:
99 × 101
Solution:
99 × 101
= (100 − 1)(100 + 1)
Using:
(a − b)(a + b) = a² − b²
= 100² − 1²
= 10000 − 1
= 9999
Example 14: Square Calculation Using Identity
Question: Find 103².
Solution:
103² = (100 + 3)²
= 100² + 3² + 2×100×3
= 10000 + 9 + 600
= 10609
Example 15: Cube Calculation Using Identity
Question: Find 11³.
Solution:
11³ = (10 + 1)³
Using:
(a + b)³ = a³ + b³ + 3ab(a + b)
= 1000 + 1 + 3×10×1×11
= 1001 + 330
= 1331
Example 16: Division Property of Square Root
Question: Simplify:
√(144/9)
Solution:
Using:
√(x/y) = √x / √y
= √144 / √9
= 12 / 3
= 4
Example 17: Perfect Square Identification
Question: Is 2025 a perfect square?
Solution:
45 × 45 = 2025
Therefore:
2025 is a perfect square.
Example 18: Perfect Cube Identification
Question: Is 4913 a perfect cube?
Solution:
17 × 17 × 17 = 4913
Therefore:
4913 is a perfect cube.
Important Exam Tips
- Memorize squares up to 50 and cubes up to 20.
- Learn unit digit patterns carefully.
- Practice prime factorization regularly.
- Use algebraic identities for fast calculations.
- Apply approximation methods in non-perfect root problems.
- Recognize perfect squares and cubes quickly.
- Practice mental calculations daily.
Practicing solved examples regularly improves calculation speed, conceptual clarity, and accuracy in solving Square Roots and Cube Roots questions in competitive examinations.