Square Root Calculator
Perfect Squares
Perfect Cubes
Common Roots
Related Calculators
Understanding Square Roots and Radicals
What is a Square Root?
The square root of a number x is a value y such that y² = x. In other words, it's the number that, when multiplied by itself, gives the original number. Every positive number has two square roots: one positive (principal) and one negative.
Square Root Definition
Example: √25 = 5 because 5 × 5 = 25
Types of Roots
Square Root (√)
The 2nd root
Cube Root (∛)
The 3rd root
Nth Root (ⁿ√)
Any root
Perfect Squares
A perfect square is a number that has an integer square root. These are the squares of whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Simplifying Square Roots
A square root can be simplified by factoring out perfect squares. The goal is to write the radical in the form a√b where b has no perfect square factors other than 1.
Simplification Steps:
More Examples:
Cube Roots and Higher
Cube roots work similarly to square roots but involve finding a number that, when multiplied by itself three times, gives the original number. Higher roots follow the same pattern.
Perfect Cubes
Fourth Powers
Negative Numbers and Roots
Important Rules
- • Even roots of negative numbers are not real (they're imaginary)
- • Odd roots of negative numbers are real and negative
- • Example: √(-16) is imaginary, but ∛(-27) = -3
Properties of Radicals
Estimating Square Roots
To estimate a square root mentally, find the two perfect squares it falls between.
Estimation Example: √50
Since 49 = 7² and 64 = 8², we know √50 is between 7 and 8.
Since 50 is closer to 49, √50 ≈ 7.07 (actual value: 7.071...)
Frequently Asked Questions
How do you simplify a square root?
Factor out perfect squares from under the radical. For example, to simplify the square root of 72: find that 72 = 36 x 2, where 36 is a perfect square. So the square root of 72 equals the square root of 36 times the square root of 2, which simplifies to 6 times the square root of 2.
Can you take the square root of a negative number?
Not in the real number system. The square root of -1 is defined as 'i' (imaginary unit) in the complex number system. So the square root of -16 equals 4i. Calculators in 'real' mode will show an error, while 'complex' mode will give the imaginary result.
What is the difference between a square root and cube root?
A square root finds what number times itself equals the input (root of 9 is 3 because 3 x 3 = 9). A cube root finds what number times itself THREE times equals the input (cube root of 27 is 3 because 3 x 3 x 3 = 27). Cube roots can be taken of negative numbers.
How do you estimate a square root without a calculator?
Find the two perfect squares the number falls between. For the square root of 50: since 49 equals 7 squared and 64 equals 8 squared, the answer is between 7 and 8. Since 50 is closer to 49, estimate around 7.07. The actual value is approximately 7.071.