Square Roots & Cube Roots: Methods, Tables & Tricks for SSC Exams

Squares and cubes to know by heart

Which numbers can be perfect squares?

• A perfect square ends only in 0, 1, 4, 5, 6 or 9. Anything ending in 2, 3, 7 or 8 is not a square.
• A square ends in an even number of zeros (100, 3,600). 1,000 cannot be a perfect square.
• A square ending in 5 ends in 25 (35² = 1,225).

Square root by the unit-digit method

Worked example: square root of 7,056. It ends in 6, so the root ends in 4 or 6. Drop the last two digits to get 70. Since 8² = 64 <= 70 < 81, the tens digit is 8. To choose between 84 and 86, compare 70 with 8 × 9 = 72: 70 is smaller, so take the smaller ending. Root = 84 (84² = 7,056).

Cube root by grouping

Cube unit digits: a root ending in 2 gives a cube ending in 8 and vice versa; 3 and 7 swap the same way; 0, 1, 4, 5, 6 and 9 keep their own digit. Worked example: cube root of 2,74,625. The last three digits end in 5, so the root ends in 5. The remaining part is 274, and 6³ = 216 <= 274 < 343 = 7³, so the tens digit is 6. Root = 65.

Prime factorisation method

Pair up the primes for a square root; make triples for a cube root. Example: 1,764 = 2² × 3² × 7², so its square root is 2 × 3 × 7 = 42.

• Smallest multiplier for a perfect square: 1,008 = 2^4 × 3² × 7. Only 7 has an odd power, so multiply by 7 to get 7,056 = 84².
• Smallest multiplier for a perfect cube: 1,372 = 2² × 7³. Multiply by 2 to get 2,744 = 14³.
• Smallest divisor: divide out the primes with leftover powers instead of multiplying.

Nearest perfect square

To find the least number to subtract from (or add to) N to get a perfect square, find the squares on either side. For 5,000: 70² = 4,900 and 71² = 5,041, so subtract 100 or add 41.

Roots of decimals

Make the number of decimal places even (for square roots) or a multiple of 3 (for cube roots), then halve or divide by 3. sqrt 0.000441 = 0.021 (six places become three). cube root of 0.008 = 0.2.

Nested and infinite roots

If x = sqrt(n + sqrt(n + ...)), then x² = n + x. For n = 6 this gives x = 3. If x = sqrt(n - sqrt(n - ...)), then x² = n - x; for n = 20, x = 4. Shortcut: for the plus form, write n as k(k - 1) and the answer is k; for the minus form, write n as k(k + 1).

Surds: simplify and rationalise

• Take out perfect-square factors: sqrt 72 = sqrt(36 × 2) = 6 sqrt 2.
• Add only like surds: sqrt 75 - sqrt 27 = 5 sqrt 3 - 3 sqrt 3 = 2 sqrt 3.
• Rationalise with the conjugate: 1/(sqrt 5 - sqrt 3) = (sqrt 5 + sqrt 3)/(5 - 3) = (sqrt 5 + sqrt 3)/2.
• Use sqrt 2 ~ 1.414, sqrt 3 ~ 1.732, sqrt 5 ~ 2.236 only at the last step.