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Master nth Root

Learn the secrets of rapid mental computation used in competitive exams. These strategies will help you bypass traditional paper-and-pencil methods.

➕ Addition ➖ Subtraction ✖️ Multiplication ➗ Division x² Squaring xⁿ Exponentiation √ nth Root % Percentage △ Pythagorean Triples lcm LCM hcf HCF

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★ Cube Roots matching last digit

For perfect cubes, the last digit of the root often matches the cube. Valid for ends 1, 4, 5, 6, 9, 0. (2/8 and 3/7 swap).

Example

∛64 ends in 4, so root ends in 4. ∛125 ends in 5, so root is 5.

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★ Square root bracketing

Find the nearest tens perfect squares above and below. The root is between them.

Example

√144 is between 10²(100) and 20²(400). It ends in 4, so root must end in 2 or 8. 12 makes sense.

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★ Factor Out Perfect Squares

If a number isn't a small perfect square, divide it by known squares (4, 9, 16, 25) to see if it reduces.

Example

√196: Can divide by 4 to get 49. √196 = √(4×49) = 2 × 7 = 14

⭐ Must Memorize: Roots of Powers of 2, 3, 4, 5

Know these perfect powers and their roots instantly — recognising them is the key to fast root calculation.

2ⁿ roots

2 = 2

√ 4 = 2

∛ 8 = 2

4√ 16 = 2

5√ 32 = 2

6√ 64 = 2

7√ 128 = 2

8√ 256 = 2

9√ 512 = 2

10√ 1024 = 2

3ⁿ roots

3 = 3

√ 9 = 3

∛ 27 = 3

4√ 81 = 3

5√ 243 = 3

6√ 729 = 3

7√ 2187 = 3

8√ 6561 = 3

9√ 19683 = 3

10√ 59049 = 3

4ⁿ roots

4 = 4

√ 16 = 4

∛ 64 = 4

4√ 256 = 4

5√ 1024 = 4

6√ 4096 = 4

7√ 16384 = 4

8√ 65536 = 4

9√ 262144 = 4

10√ 1048576 = 4

5ⁿ roots

5 = 5

√ 25 = 5

∛ 125 = 5

4√ 625 = 5

5√ 3125 = 5

6√ 15625 = 5

7√ 78125 = 5

8√ 390625 = 5

9√ 1953125 = 5

10√ 9765625 = 5

📋 Roots Reference Table (base 2–25)

Perfect squares and cubes with their square and cube roots.

base	n² → √	n³ → ∛
2	√4 = 2	∛8 = 2
3	√9 = 3	∛27 = 3
4	√16 = 4	∛64 = 4
5	√25 = 5	∛125 = 5
6	√36 = 6	∛216 = 6
7	√49 = 7	∛343 = 7
8	√64 = 8	∛512 = 8
9	√81 = 9	∛729 = 9
10	√100 = 10	∛1000 = 10
11	√121 = 11	∛1331 = 11
12	√144 = 12	∛1728 = 12
13	√169 = 13	∛2197 = 13
14	√196 = 14	∛2744 = 14
15	√225 = 15	∛3375 = 15
16	√256 = 16	∛4096 = 16
17	√289 = 17	∛4913 = 17
18	√324 = 18	∛5832 = 18
19	√361 = 19	∛6859 = 19
20	√400 = 20	∛8000 = 20
21	√441 = 21	∛9261 = 21
22	√484 = 22	∛10648 = 22
23	√529 = 23	∛12167 = 23
24	√576 = 24	∛13824 = 24
25	√625 = 25	∛15625 = 25