Squares, Cubes & Roots
Find squares, cubes, square roots and cube roots with full working — prime factorisation in pairs and triplets, the long division method step by step, and perfect-square checks.
A square root undoes a square. By prime factorisation, group the primes in pairs and take one from each pair: 324 = (2 × 2) × (3 × 3) × (3 × 3), so √324 = 2 × 3 × 3 = 18. For a cube root, group the primes in triplets. The long division method works for any number, even non-perfect squares.
What do you want to find?
Pick a mode, type a number and the working appears instantly.
Prime factorisation method
Long division method
The classic layout, one pair of digits at a time.
Estimate & check
Other ways to see it
Short-cuts and patterns from your textbook.
Tables to learn
Knowing these by heart makes roots much faster.
Patterns: unit digits
The last digit of a number decides the last digit of its square and cube.
- A perfect square never ends in 2, 3, 7 or 8.
- A perfect square ends in an even number of zeros (100, 4900, 250000).
- (n + 1)² − n² = 2n + 1, so there are 2n non-square numbers between n² and (n + 1)².
- Cubes keep the sign: (−3)³ = −27, so negative numbers have real cube roots.
From squares to roots, step by step
A square root undoes a square, a cube root undoes a cube. Here are the methods your textbook uses.
Choose a mode
Square, cube, square root or cube root — each shows the methods that suit it.
Type your number
Whole numbers, decimals and (for cubes and cube roots) negative numbers all work.
Follow the working
See prime factors grouped in pairs or triplets and the long division layout one step at a time.
Learn the tables
Print the squares (1–50) and cubes (1–20) tables for quick revision.
Prime factorisation method
Break the number into primes and group equal primes in pairs (for √) or triplets (for ∛). Take one prime from each group.
324 = (2 × 2) × (3 × 3) × (3 × 3) → √324 = 2 × 3 × 3 = 18.
If a prime is left over, the number is not a perfect square — multiply or divide by the leftover primes to make it one.
Long division method
- Pair the digits from the decimal point (left for whole numbers, right for decimals).
- Find the biggest square ≤ the first pair; write its root above and as the divisor.
- Subtract, bring down the next pair, double the quotient and find the digit x so that (double)x × x fits.
- Repeat. Add pairs of zeros to get more decimal places.
Repeated subtraction
The sum of the first n odd numbers is n². So subtract 1, 3, 5, 7 … from the number: if you reach 0 after n steps, the square root is n.
36 − 1 − 3 − 5 − 7 − 9 − 11 = 0 (6 steps) → √36 = 6.
Cube roots by estimation
For a perfect cube up to 6 digits, split it into groups of three from the right. The last group’s unit digit gives the root’s unit digit; the first group gives the tens digit.
857 | 375 → ends in 5 → unit 5; 9³ = 729 ≤ 857 < 1000 → tens 9 → ∛857375 = 95.
Squares, Cubes & Roots — your questions
How do I find a square root by prime factorisation?
Break the number into prime factors, group equal primes in pairs, and take one prime from each pair. For 7056 = 2⁴ × 3² × 7², the pairs give 2 × 2 × 3 × 7 = 84, so √7056 = 84. If a prime is left without a partner, the number is not a perfect square.
How does the long division method for square roots work?
Pair the digits from the decimal point — to the left for whole numbers, to the right for decimals. Find the largest square that fits the first pair and write its root above and as the divisor. Subtract, bring down the next pair, double the quotient so far and find the digit x where (double)x × x fits. Repeat for each pair.
How can I tell if a number is a perfect square?
A perfect square never ends in 2, 3, 7 or 8, and it always ends in an even number of zeros, as in 100, 4900 and 250000. If prime factorisation leaves every prime in a complete pair, the number is a perfect square. These checks rule out most numbers in seconds during an exam.
How do I find the cube root of a 6-digit perfect cube quickly?
Split the number into groups of three from the right. The last digit of the final group gives the unit digit of the root, and the first group gives the tens digit. For 857375, split as 857 | 375: it ends in 5 so the units digit is 5, and 9³ = 729 is the largest cube at or below 857, so the answer is 95.
What is the repeated subtraction method for square roots?
The sum of the first n odd numbers equals n², so subtract 1, 3, 5, 7 … in turn. If you reach exactly 0 after n steps, the square root is n. For 36: 36 − 1 − 3 − 5 − 7 − 9 − 11 = 0 in six steps, so √36 = 6. If you never reach zero, the number is not a perfect square.
Can a negative number have a cube root?
Yes. Cubing keeps the sign, because (−3) × (−3) × (−3) = −27, so the cube root of −27 is −3. Square roots are different: no real number squared gives a negative result, so negative numbers have no real square root. This tool accepts negative inputs for cubes and cube roots only.