Square Root Calculator
This square root calculator finds the square root of any number, plus cube roots and higher roots. For whole numbers it also gives the answer in simplified radical form, such as √72 = 6√2, alongside the decimal value.
Result
√72 ≈ 8.4852813742
| Simplified radical form | 6√2 |
|---|---|
| Check | 8.4852813742² ≈ 72 |
| As an exponent | 72^(1/2) |
Results update as you type.
How to Find and Simplify a Square Root
Decimal value
√x = x^(1/2), and in general the nth root of x = x^(1/n). Most square roots are irrational, so their decimals never end or repeat; the calculator shows up to 10 decimal places.
Simplified radical form
- Find the largest perfect square (for square roots) or perfect cube (for cube roots) that divides the number.
- Write the number as that perfect power × what’s left.
- Take the root of the perfect power and write it in front of the radical.
Worked Example
Simplify √72:
- Perfect squares that divide 72: 4, 9 and 36. The largest is 36.
- 72 = 36 × 2
- √72 = √36 × √2 = 6√2
- As a decimal: 6 × 1.41421356 = 8.4852813742
Cube root example: ∛250 = ∛(125 × 2) = 5∛2 ≈ 6.2996.
Perfect Squares Table
| Number | Square | Number | Square |
|---|---|---|---|
| 1 | 1 | 7 | 49 |
| 2 | 4 | 8 | 64 |
| 3 | 9 | 9 | 81 |
| 4 | 16 | 10 | 100 |
| 5 | 25 | 11 | 121 |
| 6 | 36 | 12 | 144 |
Tips and Common Mistakes
Roots don’t split over addition
√(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. A root can be split over multiplication, √(a × b) = √a × √b, but never over addition.
Estimate first
√72 lies between √64 = 8 and √81 = 9, so an answer of about 8.5 makes sense.
Use the largest perfect square
√72 = √(4 × 18) = 2√18 isn’t fully simplified, because 18 still contains 9. Using the largest perfect square, 36, gets there in one step.
For expressions that combine roots with other operations, use the scientific calculator.
Frequently Asked Questions
What is a square root?
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 49 is 7, because 7 × 7 = 49. Every positive number has two square roots, 7 and −7, but √49 means the positive one.
How do you simplify a square root?
Find the largest perfect square that divides the number, and take its root outside the radical. 72 = 36 × 2, and √36 = 6, so √72 = 6√2. If no perfect square other than 1 divides the number, the root is already in simplest form, like √15.
What is the square root of a negative number?
It isn't a real number, because any real number multiplied by itself is positive or zero. In complex numbers, √−1 is written i, so √−9 = 3i. Cube roots and other odd roots of negative numbers are real, though; ∛−27 = −3.
How do you find a cube root?
The cube root of a number is the value that, used three times in a multiplication, gives that number. ∛64 = 4, because 4 × 4 × 4 = 64. Enter 3 in the Root box to find one.
What are the perfect squares?
Numbers that are the square of a whole number. The first twelve are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. Their square roots are whole numbers, 1 to 12.
How is a root written as a power?
The nth root of x equals x to the power 1/n. So √x = x^(1/2) and ∛x = x^(1/3). This is how the calculator works out roots higher than 2.
Related Calculators
- Scientific CalculatorTrig, logs, powers, roots and factorials, in degrees or radians.
- Fraction CalculatorAdd, subtract, multiply and divide fractions and mixed numbers.
- GCF and LCM CalculatorGreatest common factor and least common multiple, with prime factors.
- Standard Deviation CalculatorSample and population standard deviation, variance and mean.
- Algebra CalculatorSolve linear, quadratic and cubic equations, or simplify expressions.
Sources & Assumptions
- OpenStax, Intermediate Algebra 2e — Chapter 8, Roots and Radicals (Sections 8.1 Simplify Expressions with Roots, 8.2 Simplify Radical Expressions)
- Decimal results show up to 10 decimal places. Simplified radical form is given for whole numbers up to one trillion.
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