Square Roots and Cube Roots — Practice Questions – KLEE Arithmetic & Mental Ability Test 3
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Solved examples
√53824 = ?
Answer: 232 — By the division method, √53824 = 232.
The square root of 41209 is:
Answer: 203 — By the division method, √41209 = 203.
The square root of 123454321 is:
Answer: 11111 — The pattern 1234…n…4321 is the square of n ones; here n = 5.
The number of digits in the square root of 625685746009 is:
Answer: 6 — A 12-digit perfect square has a 12/2 = 6-digit square root.
√(√17956 + √24025) = ?
Answer: 17 — √17956 = 134 and √24025 = 155; 134 + 155 = 289 and √289 = 17.
√(√44944 + √52441) = ?
Answer: 21 — √44944 = 212 and √52441 = 229; 212 + 229 = 441 and √441 = 21.
One-fourth of the sum of the prime numbers greater than 4 but less than 16 is the square of:
Answer: 3 — 5 + 7 + 11 + 13 = 36; 36/4 = 9 = 3².
√(176 + √2401) is equal to:
Answer: 15 — √2401 = 49; 176 + 49 = 225; √225 = 15.
√196/7 × √441/7 × 120/15 = ?
Answer: 48 — 14/7 = 2, 21/7 = 3, 120/15 = 8; 2 × 3 × 8 = 48.
If √x × √484 = 1034, then x = ?
Answer: 2209 — √484 = 22, so √x = 1034/22 = 47 and x = 47² = 2,209.
If √11881 × √x = 10137, then x = ?
Answer: 8649 — √11881 = 109, so √x = 10137/109 = 93 and x = 93² = 8,649.
In the equation 4050/√x = 450, the value of x is:
Answer: 81 — √x = 4050/450 = 9, so x = 81.
The square root of (272² − 128²) is:
Answer: 240 — (272 + 128)(272 − 128) = 400 × 144 = 57,600; √57,600 = 240.
√(110 + 1/4) = ?
Answer: 10.5 — 110 + 1/4 = 441/4; √(441/4) = 21/2 = 10.5.
Practice questions
√119025 = ?
- A. 335
- B. 345
- C. 355
- D. 365
Answer: Option B
Explanation:
345 × 345 = 1,19,025 (a number ending in 25 has a root ending in 5).
The square root of 10404 is:
- A. 101
- B. 112
- C. 104
- D. 102
Answer: Option D
Explanation:
102 × 102 = 10,404.
The square root of 40804 is:
- A. 222
- B. 202
- C. 212
- D. 204
Answer: Option B
Explanation:
202 × 202 = 40,804.
The square root of 91204 is:
- A. 298
- B. 302
- C. 312
- D. 304
Answer: Option B
Explanation:
302 × 302 = 91,204.
The square root of 161604 is:
- A. 402
- B. 404
- C. 412
- D. 398
Answer: Option A
Explanation:
402 × 402 = 1,61,604.
The square root of 36481 is:
- A. 189
- B. 191
- C. 199
- D. 181
Answer: Option B
Explanation:
191 × 191 = 36,481.
The square root of 12321 is:
- A. 121
- B. 111
- C. 11
- D. 1111
Answer: Option B
Explanation:
111² = 12,321 (three 1s, peak digit 3).
The square root of 1234321 is:
- A. 1,221
- B. 111
- C. 11,111
- D. 1,111
Answer: Option D
Explanation:
1111² = 12,34,321 (four 1s, peak digit 4).
The square root of 121 is:
- A. 11
- B. 12
- C. 21
- D. 111
Answer: Option A
Explanation:
11² = 121.
The square root of 12345678987654321 is:
- A. 1111111111
- B. 123456789
- C. 11111111
- D. 111111111
Answer: Option D
Explanation:
Nine 1s squared give 12345678987654321 (peak digit 9).
The square root of 1234567654321 is:
- A. 11111111
- B. 111111
- C. 1111111
- D. 1234567
Answer: Option C
Explanation:
Seven 1s squared give 1234567654321 (peak digit 7).
The number of digits in the square root of 15241383936 is:
- A. 11
- B. 5
- C. 6
- D. 7
Answer: Option C
Explanation:
It has 11 digits (odd), so the root has (11 + 1)/2 = 6 digits.
The square root of a 9-digit perfect square has how many digits?
- A. 4
- B. 9
- C. 5
- D. 3
Answer: Option C
Explanation:
n = 9 is odd: (9 + 1)/2 = 5.
The square root of a 7-digit perfect square has how many digits?
- A. 3
- B. 7
- C. 4
- D. 5
Answer: Option C
Explanation:
n = 7 is odd: (7 + 1)/2 = 4.
The square root of a 14-digit perfect square has how many digits?
- A. 8
- B. 14
- C. 7
- D. 6
Answer: Option C
Explanation:
n = 14 is even: 14/2 = 7.
The square root of a 16-digit perfect square has how many digits?
- A. 4
- B. 9
- C. 8
- D. 7
Answer: Option C
Explanation:
n = 16 is even: 16/2 = 8.
√(√1296 + √169) = ?
- A. 6
- B. 8
- C. 7
- D. 49
Answer: Option C
Explanation:
√1296 = 36, √169 = 13; 36 + 13 = 49 and √49 = 7.
√(√5184 + √81) = ?
- A. 9
- B. 11
- C. 81
- D. 8
Answer: Option A
Explanation:
√5184 = 72, √81 = 9; 72 + 9 = 81 and √81 = 9.
√(√10000 + √441) = ?
- A. 121
- B. 12
- C. 10
- D. 11
Answer: Option D
Explanation:
100 + 21 = 121 and √121 = 11.
√(√19600 + √841) = ?
- A. 169
- B. 13
- C. 12
- D. 14
Answer: Option B
Explanation:
140 + 29 = 169 and √169 = 13.
√(√40000 + √625) = ?
- A. 16
- B. 25
- C. 225
- D. 15
Answer: Option D
Explanation:
200 + 25 = 225 and √225 = 15.
√(√3136 + √64) = ?
- A. 64
- B. 8
- C. 7
- D. 9
Answer: Option B
Explanation:
√3136 = 56, √64 = 8; 56 + 8 = 64 and √64 = 8.
√(√8836 + √36) = ?
- A. 100
- B. 10
- C. 9
- D. 11
Answer: Option B
Explanation:
√8836 = 94, √36 = 6; 94 + 6 = 100 and √100 = 10.
√(√12100 + √1156) = ?
- A. 11
- B. 13
- C. 12
- D. 144
Answer: Option C
Explanation:
110 + 34 = 144 and √144 = 12.
√(√32400 + √256) = ?
- A. 196
- B. 13
- C. 14
- D. 16
Answer: Option C
Explanation:
180 + 16 = 196 and √196 = 14.
√(√62500 + √1521) = ?
- A. 289
- B. 16
- C. 17
- D. 19
Answer: Option C
Explanation:
250 + 39 = 289 and √289 = 17.
The sum of all prime numbers less than 10 is:
- A. 18
- B. 28
- C. 17
- D. 15
Answer: Option C
Explanation:
2 + 3 + 5 + 7 = 17.
The sum of the prime numbers between 1 and 20 is:
- A. 75
- B. 79
- C. 60
- D. 77
Answer: Option D
Explanation:
2+3+5+7+11+13+17+19 = 77.
The sum of the prime numbers between 30 and 50 is:
- A. 201
- B. 160
- C. 189
- D. 199
Answer: Option D
Explanation:
31 + 37 + 41 + 43 + 47 = 199.
The sum of the prime numbers between 40 and 60 is the fifth power of:
- A. 3
- B. 4
- C. 2
- D. 5
Answer: Option A
Explanation:
41 + 43 + 47 + 53 + 59 = 243 = 3⁵.
How many prime numbers are there between 1 and 50?
- A. 25
- B. 15
- C. 14
- D. 16
Answer: Option B
Explanation:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 — 15 primes.
√(100 + √441) is equal to:
- A. 10
- B. 11
- C. 12
- D. 21
Answer: Option B
Explanation:
√441 = 21; 100 + 21 = 121; √121 = 11.
√(161 + √1225) is equal to:
- A. 13
- B. 35
- C. 14
- D. 15
Answer: Option C
Explanation:
√1225 = 35; 161 + 35 = 196; √196 = 14.
√(230 + √3481) is equal to:
- A. 16
- B. 18
- C. 17
- D. 59
Answer: Option C
Explanation:
√3481 = 59; 230 + 59 = 289; √289 = 17.
√(300 + √576) is equal to:
- A. 18
- B. 24
- C. 19
- D. 17
Answer: Option A
Explanation:
√576 = 24; 300 + 24 = 324; √324 = 18.
√(52 + √144) is equal to:
- A. 9
- B. 7
- C. 8
- D. 12
Answer: Option C
Explanation:
√144 = 12; 52 + 12 = 64; √64 = 8.
√256/8 × √324/9 × 60/12 = ?
- A. 20
- B. 24
- C. 40
- D. 16
Answer: Option A
Explanation:
16/8 = 2, 18/9 = 2, 60/12 = 5; 2 × 2 × 5 = 20.
√625/5 × √729/9 × 72/24 = ?
- A. 45
- B. 15
- C. 27
- D. 75
Answer: Option A
Explanation:
25/5 = 5, 27/9 = 3, 72/24 = 3; 5 × 3 × 3 = 45.
√484/11 × √361/19 × 90/15 = ?
- A. 6
- B. 12
- C. 18
- D. 24
Answer: Option B
Explanation:
22/11 = 2, 19/19 = 1, 90/15 = 6; 2 × 1 × 6 = 12.
√1024/16 × √900/10 × 49/7 = ?
- A. 42
- B. 36
- C. 21
- D. 48
Answer: Option A
Explanation:
32/16 = 2, 30/10 = 3, 49/7 = 7; 2 × 3 × 7 = 42.
√169/13 × √576/8 × 150/25 = ?
- A. 36
- B. 24
- C. 12
- D. 18
Answer: Option D
Explanation:
13/13 = 1, 24/8 = 3, 150/25 = 6; 1 × 3 × 6 = 18.
If √x × √361 = 703, then x = ?
- A. 1,296
- B. 1,369
- C. 1,444
- D. 37
Answer: Option B
Explanation:
√361 = 19, so √x = 703/19 = 37 and x = 37² = 1,369.
If √x × √576 = 1032, then x = ?
- A. 43
- B. 1,936
- C. 1,849
- D. 1,764
Answer: Option C
Explanation:
√576 = 24, so √x = 1032/24 = 43 and x = 43² = 1,849.
If √x × √289 = 884, then x = ?
- A. 2,809
- B. 2,704
- C. 52
- D. 2,601
Answer: Option B
Explanation:
√289 = 17, so √x = 884/17 = 52 and x = 52² = 2,704.
If √x × √729 = 1647, then x = ?
- A. 61
- B. 3,721
- C. 3,600
- D. 3,844
Answer: Option B
Explanation:
√729 = 27, so √x = 1647/27 = 61 and x = 61² = 3,721.
If √x × √196 = 532, then x = ?
- A. 38
- B. 1,369
- C. 1,444
- D. 1,521
Answer: Option C
Explanation:
√196 = 14, so √x = 532/14 = 38 and x = 38² = 1,444.
If √12769 × √x = 9153, then x = ?
- A. 6,561
- B. 81
- C. 6,724
- D. 6,400
Answer: Option A
Explanation:
√12769 = 113, so √x = 9153/113 = 81 and x = 81² = 6,561.
If √10609 × √x = 7004, then x = ?
- A. 68
- B. 4,761
- C. 4,489
- D. 4,624
Answer: Option D
Explanation:
√10609 = 103, so √x = 7004/103 = 68 and x = 68² = 4,624.
If √16129 × √x = 11049, then x = ?
- A. 7,744
- B. 87
- C. 7,569
- D. 7,396
Answer: Option C
Explanation:
√16129 = 127, so √x = 11049/127 = 87 and x = 87² = 7,569.
If √14641 × √x = 8833, then x = ?
- A. 73
- B. 5,329
- C. 5,184
- D. 5,476
Answer: Option B
Explanation:
√14641 = 121, so √x = 8833/121 = 73 and x = 73² = 5,329.
If √22801 × √x = 14345, then x = ?
- A. 9,025
- B. 9,216
- C. 95
- D. 8,836
Answer: Option A
Explanation:
√22801 = 151, so √x = 14345/151 = 95 and x = 95² = 9,025.
In the equation 3600/√x = 300, the value of x is:
- A. 121
- B. 12
- C. 169
- D. 144
Answer: Option D
Explanation:
√x = 3600/300 = 12, so x = 144.
In the equation 5625/√x = 375, the value of x is:
- A. 225
- B. 15
- C. 256
- D. 196
Answer: Option A
Explanation:
√x = 5625/375 = 15, so x = 225.
In the equation 2304/√x = 144, the value of x is:
- A. 16
- B. 289
- C. 225
- D. 256
Answer: Option D
Explanation:
√x = 2304/144 = 16, so x = 256.
In the equation 7056/√x = 336, the value of x is:
- A. 21
- B. 484
- C. 400
- D. 441
Answer: Option D
Explanation:
√x = 7056/336 = 21, so x = 441.
In the equation 1890/√x = 105, the value of x is:
- A. 324
- B. 361
- C. 289
- D. 18
Answer: Option A
Explanation:
√x = 1890/105 = 18, so x = 324.
The square root of (65² − 63²) is:
- A. 8
- B. 16
- C. 2
- D. 32
Answer: Option B
Explanation:
(65 + 63)(65 − 63) = 128 × 2 = 256; √256 = 16.
The square root of (17² − 8²) is:
- A. 9
- B. 25
- C. 13
- D. 15
Answer: Option D
Explanation:
(17 + 8)(17 − 8) = 25 × 9 = 225; √225 = 15.
The square root of (125² − 35²) is:
- A. 120
- B. 100
- C. 160
- D. 90
Answer: Option A
Explanation:
(125 + 35)(125 − 35) = 160 × 90 = 14,400; √14,400 = 120.
The square root of (41² − 9²) is:
- A. 50
- B. 40
- C. 32
- D. 36
Answer: Option B
Explanation:
(41 + 9)(41 − 9) = 50 × 32 = 1,600; √1,600 = 40.
The square root of (145² − 143²) is:
- A. 24
- B. 2
- C. 48
- D. 12
Answer: Option A
Explanation:
(145 + 143)(145 − 143) = 288 × 2 = 576; √576 = 24.
√(56 + 1/4) = ?
- A. 7.5
- B. 7.25
- C. 8.5
- D. 7.05
Answer: Option A
Explanation:
56 + 1/4 = 225/4; √(225/4) = 15/2 = 7.5.
√(30 + 1/4) = ?
- A. 5.5
- B. 6.5
- C. 5.25
- D. 5.05
Answer: Option A
Explanation:
30 + 1/4 = 121/4; √(121/4) = 11/2 = 5.5.
√(12 + 1/4) = ?
- A. 3.25
- B. 3.05
- C. 3.5
- D. 4.5
Answer: Option C
Explanation:
12 + 1/4 = 49/4; √(49/4) = 7/2 = 3.5.
√(2 + 7/9) = ?
- A. 1 7/9
- B. 2 1/3
- C. 1 1/3
- D. 1 2/3
Answer: Option D
Explanation:
2 + 7/9 = 25/9; √(25/9) = 5/3 = 1 2/3.
√(1 + 9/16) = ?
- A. 1.75
- B. 1.5
- C. 1.125
- D. 1.25
Answer: Option D
Explanation:
1 + 9/16 = 25/16; √(25/16) = 5/4 = 1.25.