Square Root Calculator: Square Roots, Cube Roots and Simplest Radical Form
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The square root of 50 is about 7.071068. In simplest radical form, √50 = 5√2.
The square root of 50 is 7.071068. In simplest radical form it is 5√2.
Nearby perfect squares
| Number | Number to the power 2 |
|---|---|
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
Show the steps
- Prime factorization: 50 = 2 × 5^2.
- Pull out each prime that appears 2 times: 50 = 5^2 × 2, so the root is 5√2.
- Decimal value: 7.0710678119, because 7.0710678119^2 is about 50.0000.
Roots
The n-th root of x is the number r with r^n = x
√(a × b) = √a × √b (for a, b at least 0)
√(a^2 × b) = a√b
- The square root of a negative number is not a real number. It is written with i, where i × i = −1.
Estimates for planning and learning, not financial, tax, legal or medical advice. Check figures with your lender, tax preparer or doctor before you act.
About this calculator
The square root of a number is the value that gives the number when you multiply it by itself. The square root of 49 is 7 because 7 × 7 = 49. This calculator gives the root as a decimal, tells you whether the number is a perfect square, and writes whole numbers in simplest radical form, such as √72 = 6√2.
You can also switch to a cube root, a 4th root or a 5th root, enter a decimal, or use a negative number to see an imaginary result.
How to use it
- Type a number. Decimals and negative numbers are allowed.
- Choose square root, cube root, 4th root or 5th root.
- Read the decimal and the simplest radical form. Open Show the steps to see the prime factor method.
How to simplify a square root
Factor the number into primes. Every pair of equal primes comes out of the root as one. For 72 = 2 × 2 × 2 × 3 × 3, one pair of 2s and one pair of 3s come out, giving 2 × 3 = 6. What is left inside is 2, so √72 = 6√2.
Perfect squares to know
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and 225 are the squares of 1 to 15. A number that is not on a list like this has an irrational square root: its decimal never ends or repeats.
Negative numbers
No real number squared is negative, so the square root of a negative number is imaginary. √(−25) = 5i. An odd root, such as a cube root, of a negative number is a negative real number: the cube root of −8 is −2.
Sources
Quick check: test yourself
What is √169?
Correct answer: 13. 13 × 13 = 169.
Simplify √72.
Correct answer: 6√2. 72 = 36 × 2, and √36 = 6.
What is the square root of −25?
Correct answer: 5i. √(−25) = 5i, where i × i = −1.
Pick an answer for each question.
Common questions
How do I find a square root by hand?
Estimate between two perfect squares, then refine. For √50, 7 × 7 = 49 and 8 × 8 = 64, so the root is just over 7: about 7.07.
What is the square root of 2?
About 1.41421356. It is irrational, so the decimal never ends.
Can you take the square root of a negative number?
Not as a real number. The result is imaginary: √(−16) = 4i.
What does simplest radical form mean?
The number under the root has no perfect square factor left. √12 simplifies to 2√3.
How is a cube root different?
A cube root is the number that gives the original when multiplied by itself three times. The cube root of 27 is 3.
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