Squares, cubes and roots in seconds

Almost every square you need is one of four patterns: ends in 5, near 50, near 100, or (a+b)². Roots of perfect squares and cubes come from the last digit plus the leading block.

Start with this

Squares and roots appear inside mensuration, number system, DI and simplification. Knowing four square patterns and one root trick saves a minute every time.

Questions like 'square root of 7744' or 'value of 113²' are asked directly, and squares hide inside Pythagoras, areas and compound interest. Learn squares to 30 by heart (cheat sheet) and use the patterns below for everything else.

Squares ending in 5

For a number n5 (like 35, 85, 115), multiply n by the next number, then write 25 after it.

Why: (10n + 5)² = 100 n² + 100 n + 25 = 100 n (n + 1) + 25.

Worked example

85²

1. 8 × 9 = 72.
   Why: n times the next number.
2. Write 25 after: 7225.

Answer: 7225

Worked example

115²

1. 11 × 12 = 132.
   Why: n can have two digits.
2. 13225.

Answer: 13225

Squares near 50 and near 100

Near 50: for 50 + k, the first part is 25 + k and the last two digits are k². 53² = 28 | 09 = 2809. 47² = 22 | 09 = 2209.

Near 100: for 100 + k, the first part is 100 + 2k and the last two digits are k². 96² = 92 | 16 = 9216. 108² = 116 | 64 = 11664.

If k² has more than two digits, carry: 113² = 126 | 169 → 127 | 69 = 12769.

Worked example

57²

1. k = 7. First part 25 + 7 = 32.
   Why: Near 50 rule.
2. Last two digits 7² = 49.
3. 3249.

Answer: 3249

Worked example

93²

1. k = -7. First part 100 - 14 = 86.
   Why: Near 100 rule; 2k = -14.
2. Last two digits 49.
3. 8649.

Answer: 8649

Any 2-digit square: a² | 2ab | b²

For a two-digit number ab, write three blocks: a², then 2 × a × b, then b², and carry from right to left, one digit per block (except the left one).

67²: 36 | 84 | 49 → write 9 carry 4; 84 + 4 = 88, write 8 carry 8; 36 + 8 = 44. Answer 4489.

Worked example

73²

1. Blocks: 49 | 42 | 9.
   Why: a² = 49, 2ab = 42, b² = 9.
2. 9 stays. 42: write 2, carry 4. 49 + 4 = 53.
3. 5329.

Answer: 5329

Square roots of perfect squares

Step 1: the last digit of the square tells you two possible last digits of the root. 1 → 1 or 9, 4 → 2 or 8, 9 → 3 or 7, 6 → 4 or 6, 5 → 5, 0 → 0.

Step 2: cover the last two digits. The number left tells you the tens digit: find the biggest square below it.

Step 3: choose between the two candidates by comparing with the square of (tens digit)5.

Square roots of perfect squares: reference

Worked example

Square root of 7744

1. Ends in 4, so the root ends in 2 or 8.
   Why: 2² = 4 and 8² = 64 both end in 4.
2. Cover 44: 77 remains. 8² = 64 ≤ 77 < 81, so the tens digit is 8.
   Why: The root is in the 80s.
3. 85² = 7225, which is less than 7744, so take the larger candidate: 88.
   Why: Above the halfway square means the bigger ending.

Answer: 88

Cube roots of perfect cubes

Last digits of cubes are all different, so the last digit of a cube gives the last digit of the root directly. Only 2 <→ 8 and 3 <→ 7 swap; every other digit stays the same.

Cover the last three digits. The number left tells you the first digit: the biggest cube below it.

Cube roots of perfect cubes: reference

Worked example

Cube root of 175616

1. Ends in 6, so the root ends in 6.
   Why: 6³ = 216 ends in 6.
2. Cover 616: 175 remains. 5³ = 125 ≤ 175 < 216, so the first digit is 5.
3. 56.

Answer: 56

Why 35² = 3 × 4 | 25 (areas)

85² = 8 × 9 | 25

Shortcut: Square root of a 4-digit perfect square

Use it when: e.g. root of 7744.

1. Last digit 4 → root ends 2 or 8.
2. Front block 77 sits between 64 (8²) and 81 (9²) → tens digit 8.
3. Choose between 82 and 88: 85² = 7225 < 7744, so the larger one.

Square root of a 4-digit perfect square: try it

Square root of 7744

1. Ends 2/8, tens 8, above 85².

Answer: 88

Shortcut: Cube root of a 6-digit perfect cube

Use it when: e.g. cube root of 175616.

1. Last digit 6 → root ends 6.
2. Strike the last three digits: 175. 5³ = 125 ≤ 175 < 216 = 6³, so tens digit 5.

Cube root of a 6-digit perfect cube: try it

Cube root of 175616

1. 56.

Answer: 56

Your turn medium

93²

1. 100 - 14 = 86 | 49.

Answer: 8649

Your turn medium

113²

1. 100 + 26 = 126 | 169 → 126 + 1 carry = 127 | 69.

Answer: 12,769

Your turn medium

Square root of 9409

1. Ends 9 → 3 or 7.
2. 94 between 81 and 100 → 9.
3. 95² = 9025 < 9409 → 97.

Answer: 97

Your turn medium

Square root of 5329

1. Ends 9 → 3/7.
2. 53 between 49 and 64 → 7.
3. 75² = 5625 > 5329 → 73.

Answer: 73

Recap

• square any number ending in 5, near 50 or near 100 in your head
• find the square root of a 4-digit perfect square in 15 seconds
• find the cube root of a 6-digit perfect cube in 10 seconds

Now practise

17 questions with full solutions.

Open practice Back to the lesson