Practice Quantitative Aptitude questions on Square Roots and Cube Roots, covering perfect squares and cubes, simplification of radicals, root calculations, estimation, and problems involving squares, cubes, and their properties. Takes about 20–25 minutes and is useful for placement tests, TCS NQT, SSC, banking and MBA entrance exams.
Question 1
What should come in place of question mark in
Find the missing value (cube root equation).
A
16✓✗
B
4✓✗
C
64✓✗
D
256✓✗
Explanation
Let ? = x. Then x1/3 / 2.56 = 100 / x, so x4/3 = 256.
x = 2563/4 = (44)3/4 = 43 = 64.
Question 2
Find the value of the nested root of 6.
A
2✓✗
B
5✓✗
C
4✓✗
D
3✓✗
Explanation
Let x = the given nested root. Then x = √6 + x, so x2 - x - 6 = 0.
(x - 3)(x + 2) = 0, and x is positive, so x = 3.
Question 3
Find the value of the nested root of 56.
A
7.48….✓✗
B
7✓✗
C
8✓✗
D
None of these✓✗
Explanation
Let x = the given nested root. Then x = √56 + x, so x2 - x - 56 = 0.
(x - 8)(x + 7) = 0, and x is positive, so x = 8.
Question 4
Find the value (given the square root of 72).
A
18.485✓✗
B
8.485✓✗
C
1.072✓✗
D
1.060✓✗
Explanation
98 = 7264, so the square root = √72 ÷ 8 = 8.485 ÷ 8 = 1.0606, which is 1.060 (nearly).
Question 5
(8.01)2 + ? = (8.97)2 What will approximately come in place of question mark?
A
13✓✗
B
14✓✗
C
19✓✗
D
16✓✗
Explanation
? = (8.97)² - (8.01)² ≈ 80.5 - 64.2 = 16.3.
So ? is approximately 16.
Question 6
Four-fifth of one-eighth of 34th of A is 64. What is the cube root of 35th of A?
A
5✓✗
B
8✓✗
C
3✓✗
D
4✓✗
Explanation
45 × 18 × 34 × A = 64, so 340 × A = 64.
35 of A = 3A40 × 8 = 64 × 8 = 512.
The cube root of 512 = 8.
Question 7
What is smallest number with which 5400 may be multiplied so that the product is perfect cube?
A
5✓✗
B
3✓✗
C
4✓✗
D
6✓✗
Explanation
5400 = 2³ × 3³ × 5². For a perfect cube, every prime must have a power that is a multiple of 3.
The power of 5 is 2, so multiply by 5 to make it 5³.
The smallest number = 5.
Question 8
What least number should be multiplied with 384 to make it a perfect square?
A
3✓✗
B
6✓✗
C
2✓✗
D
8✓✗
Explanation
384 = 2⁷ × 3. For a perfect square every prime must have an even power, so multiply by 2 × 3 = 6.
384 × 6 = 2304 = 48², so the least number is 6.
Question 9
What is the value of the sixth root of 0.004096?
A
0.4✓✗
B
4✓✗
C
0.04✓✗
D
64✓✗
Explanation
0.004096 = 0.46, because 46 = 4096 and 0.46 has 6 decimal places.
So the sixth root of 0.004096 = 0.4.
Question 10
How many two-digit numbers satisfy the following property given below:
The last digit (unit's digit) of the square of the two-digit number is 8?
A
4✓✗
B
12✓✗
C
15✓✗
D
None of these✓✗
Explanation
The last digit of a square can only be 0, 1, 4, 5, 6 or 9.
The last digit 8 is impossible, so no two-digit number satisfies the property (the answer is none of the given options).
Question 11
Given that √12 = 3.464 and √120 = 10.95, find the value of √1.2 + √1200 + √0.012.
A
32.164✓✗
B
35.844✓✗
C
36.164✓✗
D
37.304✓✗
Explanation
√1.2 = √120/10 = 1.095, √1200 = 10 × √12 = 34.64 and √0.012 = √120/100 = 0.1095.
Sum = 1.095 + 34.64 + 0.1095 = 35.8445, which is 35.844 (approximately).
Question 12
Largest five digit number which is a perfect square is
A
99225✓✗
B
99856✓✗
C
99596✓✗
D
99899✓✗
Explanation
3162 = 99856 (3172=100489 has 6 digits).
Question 13
Find the value of √0.324 × 4.624 / √1.5625 × 0.0289 × 64
A
0.64✓✗
B
0.70✓✗
C
0.72✓✗
D
0.54✓✗
Explanation
Numerator: √0.324 × 4.624=√1.498≈1.224. Denominator: √1.5625 × 0.0289 × 64=√2.89=1.7. Ratio ≈1.2241.7≈0.72.
Question 14
9n = 531441. Find n.
A
4✓✗
B
5✓✗
C
6✓✗
D
7✓✗
Explanation
9=32, 531441=312. So (32)n=312, giving 2n=12, n=6.
Question 15
√7921 / √31684 = ?
A
0.50✓✗
B
0.25✓✗
C
0.45✓✗
D
0.20✓✗
Explanation
√7921=89, √31684=178. 89178 = 0.5 (since 178 = 2 × 89).
Question 16
The smallest number added to 622517 to make the sum a perfect square is
A
2✓✗
B
3✓✗
C
4✓✗
D
5✓✗
Explanation
7892 = 622521. 622521 - 622517 = 4.
Question 17
If x=2+√3 and y=2-√3, find the value of x2+y2
A
14✓✗
B
17✓✗
C
16✓✗
D
15✓✗
Explanation
x2=7+4√3, y2=7-4√3. Sum = 14.
Question 18
Find n, given 8n=32768
A
4✓✗
B
5✓✗
C
6✓✗
D
7✓✗
Explanation
8=23, 32768=215. So (23)n=215, giving 3n=15, n=5.
Question 19
Evaluate √369.7929
A
18.13✓✗
B
17.73✓✗
C
19.23✓✗
D
23.13✓✗
Explanation
19.232 = 369.7929.
Question 20
Find the value of √0.324 / √0.00729
A
203✓✗
B
149✓✗
C
187✓✗
D
275✓✗
Explanation
√0.324 / √0.00729 = √0.3240.00729 = √4009 = 20/3.
Question 21
If x=1+√3 and y=1-√3, find the value of x2+y2
A
8✓✗
B
7✓✗
C
6✓✗
D
5✓✗
Explanation
x2=4+2√3, y2=4-2√3. Sum = 8.
Question 22
What is the least number that should be subtracted from 884740 to obtain a perfect cube?
A
8✓✗
B
9✓✗
C
4✓✗
D
7✓✗
Explanation
963 = 884736. 884740 - 884736 = 4.
Question 23
Find the square root of 2319529
A
1523✓✗
B
1433✓✗
C
1423✓✗
D
1353✓✗
Explanation
15232 = 2319529.
Question 24
Evaluate √7921
A
79✓✗
B
89✓✗
C
99✓✗
D
69✓✗
Explanation
892 = 7921.
Question 25
If x2=16, then x=?
A
+4✓✗
B
-4✓✗
C
+4 and -4✓✗
D
None✓✗
Explanation
Both +4 and -4 satisfy x2=16, since (-4)2=16 as well.
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