Square Root and Cube Root Calculator

Enter a number to get its square root or cube root as a decimal to the places you choose, and in simplest surd form — √72 = 6√2, ∛54 = 3∛2. The working shows the prime factorisation method and the long division method from NCERT Class 8, and tells you whether the number is a perfect square or cube.

Number

Whole numbers or decimals, e.g. 72 or 2.25

Decimal places

√72

≈ 8.4853

Simplest form

6√2

Not a perfect square: the decimal goes on forever, so it is rounded.

Step-by-step solution

Prime factorisation: 72 = 2³ × 3²
Make pairs of equal factors: 1 pair of 2, 1 pair of 3 — each pair comes out of the root as one factor
√72 = √(6² × 2) = 6√2
Long division method — digit groups: 72 . 00 00 00 00
Largest square ≤ 72: 8² = 64 → first digit 8, remainder 8
Bring down 00 → 800. Divisor 16_: 164 × 4 = 656 ≤ 800 → next digit 4, remainder 144
Bring down 00 → 14400. Divisor 168_: 1688 × 8 = 13504 ≤ 14400 → next digit 8, remainder 896
Bring down 00 → 89600. Divisor 1696_: 16965 × 5 = 84825 ≤ 89600 → next digit 5, remainder 4775
…repeat for the remaining 1 group (a decimal point goes in the root when you reach the decimal groups).
√72 ≈ 8.4853 (rounded to 4 decimal places)

Perfect squares 1² to 30²

1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
26² = 676
27² = 729
28² = 784
29² = 841
30² = 900

How to use the square root calculator

Choose Square root or Cube root.
Type the number — whole numbers or decimals such as 2.25; cube roots accept negative numbers.
Choose how many decimal places you want (0–10).
Read the decimal answer, the simplest surd form and the step-by-step working.

Formulas

Square root	`√n = x means x × x = n`
Cube root	`∛n = x means x × x × x = n`
Simplifying a surd	`√(a² × b) = a√b` Pull out every pair of equal prime factors: √72 = √(2 × 2 × 2 × 3 × 3) = 2 × 3 √2 = 6√2.
Product rule	`√(ab) = √a × √b, ∛(ab) = ∛a × ∛b`

Solved examples

Example 1: Find √72 in simplest form.

72 = 2³ × 3²
Pairs: one pair of 2s, one pair of 3s → 2 × 3 = 6 come out
√72 = 6√2 ≈ 6 × 1.4142

Answer: 6√2 ≈ 8.4853

Example 2: Find the square root of 1764 by prime factorisation.

1764 = 2² × 3² × 7²
Take one factor from each pair: 2 × 3 × 7

Answer: 42

Example 3: Find ∛54.

54 = 2 × 3³
One triple of 3s comes out as 3
∛54 = 3∛2

Answer: 3∛2 ≈ 3.7798

Perfect squares and how to spot them

A perfect square is the square of a whole number: 1, 4, 9, 16, 25… In its prime factorisation every prime appears an even number of times. A perfect square never ends in 2, 3, 7 or 8, and if it ends in 0 it ends in an even number of zeros. These tests let you rule out numbers quickly in exams.

The long division method

For large numbers or decimals, NCERT Class 8 uses long division: pair the digits from the decimal point, find the largest square below the first group, then repeatedly bring down the next pair and find the next digit with the "double the root and append a digit" rule. The calculator lists the first rounds so you can follow the pattern.

🎯 Asked in competitive exams

This topic regularly appears in the quantitative aptitude section of:

SSC CGL & CHSL
RRB NTPC & Group D
IBPS Clerk
Campus placement aptitude tests

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Frequently asked questions

How do I simplify a square root?

Factorise the number into primes and take one factor out for each pair. √72 = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2. Whatever has no pair stays inside the root.

What is the square root of 2?

√2 ≈ 1.41421356. It is irrational: its decimal never ends or repeats, so it cannot be written as a fraction.

Can a negative number have a square root?

Not among real numbers, because any real number squared is zero or positive. (In complex numbers, √−4 = 2i.) A negative number does have a real cube root: ∛−27 = −3.

How do I find the cube root by prime factorisation?

Factorise the number and make groups of three equal primes. Take one prime from each group. 1728 = 2⁶ × 3³ → (2 × 2) × 3 = 12, so ∛1728 = 12.

Which numbers are perfect squares from 1 to 100?

1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 — the squares of 1 to 10. The table on this page lists perfect squares and cubes up to 30.