Surds Simplification Calculator
This surds simplification calculator takes any whole number entered under a square root sign and reduces it to its simplest surd form, written as a times the square root of b, where b keeps no perfect square factors. Surds are irrational roots, numbers like the square root of 2 or 3 that never resolve into an exact fraction or whole number, and simplifying them keeps working exact rather than rounding to a decimal too early, which matters throughout algebra, trigonometry and NCEA-level maths. Type a single number into the field, such as 72, and the calculator instantly factors it into a perfect square multiplied by a remainder, pulling the square root of the perfect square out the front. You get three results: the full simplified expression, for example the square root of 72 equals 6 root 2; the decimal value of the original root, rounded for display; and the same answer broken out as a times the square root of b, so you can see each part separately. If the number you enter is itself a perfect square, such as 49, the calculator simply returns the whole number, since there is no surd left to simplify. Use it to check homework, verify working by hand, or quickly convert an unwieldy root into its neatest exact form. A worked example and the underlying method are set out below the results.
The formula
Factor the number under the root into a perfect square times a remainder: n = a squared times b, where b has no square factors. Then the square root of n equals a times the square root of b.
Worked example
72 = 36 times 2, and 36 is a perfect square, so the square root of 72 is 6 root 2, about 8.485281. Enter 72 to confirm.
Frequently asked questions
What is a surd?
An irrational root, such as the square root of 2, that cannot be written exactly as a fraction or whole number.
How do I simplify a square root?
Take out the largest perfect square factor. The square root of 50 is 5 root 2 because 50 = 25 times 2.
What if the number is a perfect square?
Then the root is a whole number and there is no surd left, for example the square root of 49 is 7.
Who this calculator is for
This calculator is for students, programmers and anyone needing a quick, reliable result.
What this calculator assumes
- You enter valid input.
- The standard algorithm is applied.
- Results are rounded for display.
Formula and sources
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