Solved Examples: Square Roots and Cube Roots | Quantitative Aptitude for SSC CGL PDF Download

Square Root

When y = x² then square root of y is x and it is written as √y = x.
For example, √9 = 3, √16 = 4 and so on.

Cube Root

When y = x³ then cube root of y is x and it is written as ∛y = x.
For example, ∛8 = 2, ∛27 = 3 and so on.

Important Formulaes

• √ab = √a x √b
• √(a/b) = √a / √b

Solved Examples

Example 1: What is the least number required to multiply to 9720 to make a perfact cube?
(a) 55
(b) 65
(c) 75
(d) 85
Ans: (c)
9261 = 2 x 2 x 2 x 3 x 3 x 3 x 5 x 3 x 3
= 2³ x 3³ x 3² x 5
∴ required no: 3 x 5² = 3 x 25 = 75

Example 2: If √ 15 = 3.8729 then what is (√5 + √3)/(√5 - √3)?
(a) 7.8729
(b) 6.8729
(c) 5.8729
(d) 4.8729
Ans: (a)
(√5 + √3)/(√5 - √3)
= (√5 + √3)/(√5 - √3) * (√5 + √3)/(√5 + √3)
= (√5 + √3)2/(5 - 3)
= (5 + 3 + 2√5 + √3)/2
= (8 + 2√15)/2
= 2(4 + √15)/2
= 4 + √15
= 4 + 3.8729
= 7.8729

Example 3: If √ 1369 = 37 then what is √13.69 + √0.1369 + √ 0.001369 + √ 0.00001369?
(a) 4.0021
(b) 4.1107
(c) 3.1232
(d) 2.1323
Ans: (b)
√13.69 + √0.1369 + √ 0.001369 + √ 0.00001369
= √(1369 / 100) + √(1369/ 10000) + √(1369/ 1000000) + √(1369/ 100000000)
= √1369 / √100 + √1369 / √10000 + √1369 / √1000000 + √1369 / √100000000
= 37/10 + 37/100 + 37/1000 + 37/ 10000
= 3.7 + 0.37 + 0.037 + 0.0037
= 4.1107

Example 4: √(25/16) = ?
(a) 3/4
(b) 5/4
(c) 4
(d) 4/5
Ans: (b)
√(25/16) = √25/ √ 16 = 5/4.

Example 5: √ 54 x √ 6 = ?
(a) 24
(b) 15
(d) 18
(e) 13
Ans: (c)
√ 54 x √ 6 = √ 54 x 6 = √ (3 x 3 x 6 x 6)
= √ (3² x 6²)
= √ 18

Example 6: ∛9261 = ?
(a) 21
(b) 17
(c) 29
(d) 23
Ans: (a)
9261 = 3 x 3 x 3 x 7 x 7 x 7
= 3³ x 7³
∴ ∛9261 = (3³ x 7³)^1/3
= 3 x 7 = 21.

Example 7: If √15 = 3.88. What is √(5/3)?
(a) 1.213
(b) 1.293
(c) 1.321
(d) 1.432
Ans: (b)
√(5/3) = (√5 / √ 3) x (√3 / √ 3) =  √15 / 3 = 3.88/ 3 = 1.293

Example 8: √(248 + √(51 + √169)) = ?
(a) 15
(b) 12
(c) 13
(d) 16
Ans: (d)
√(248 + √(51 + √169)) = √(248 + √(51 + 13))
= √(248 + √(64)) = √(248 + 8) = √(256) = 16

Example 9: √ 2025 = ?
(a) 45
(b) 35
(c) 34
(d) 30
Ans: (a)
Resolve 2025 into prime numbers first.
2025 = 5 x 5 x 3 x 3 x 3 x 3 = 5² x 3² x 3²
∴ √2025 = 5 x 3 x 3 = 45

Example 10: √64009 = ?
(a) 803
(b) 363
(c) 253
(d) 347
Ans: (c)
√64009 = √(253 x 253) = 253