Simplify Radicals and Square Roots Calculator
Enter a whole number and a root and get it in simplest radical form, such as the square root of 72 becoming 6 times the square root of 2. The prime factors are shown so you can follow the steps, negative numbers are handled, and a second tab rationalizes a fraction with a square root in the denominator. Everything runs in your browser.
72 = 2^3 x 3^2. Every prime that appears 2 times (or more) comes out from under the root: its power divided by 2, rounded down, goes in front, and the remainder stays inside.
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To simplify a root, split the number into prime factors and pull out every group of identical factors as large as the root, so a square root takes pairs and a cube root takes triples. The square root of 72 is 2 x 2 x 2 x 3 x 3 under the root sign: one pair of 2s and one pair of 3s come out as 2 x 3 = 6, and a single 2 stays, giving 6 times the square root of 2. A fraction with a root in the bottom is rationalized by multiplying top and bottom by that root. Whole numbers up to 1,000,000,000,000 are supported, and everything is worked out in your browser.
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Student ToolsAbout the Simplify Radicals and Square Roots Calculator
To simplify a radical, write the number under the root sign as a product of prime factors, then pull out every group of identical factors as large as the root. For a square root, pairs of primes come out as one. The number 72 is 2 x 2 x 2 x 3 x 3: one pair of 2s and one pair of 3s come out as 2 x 3 = 6, and a single 2 is left under the root, so the square root of 72 is 6 times the square root of 2. For a cube root, triples come out, so the cube root of 54 = 2 x 3 x 3 x 3 is 3 times the cube root of 2. The fourth root of 48 is 2 times the fourth root of 3. A root is already in simplest form when no prime under it appears as often as the root, as with the square root of 30. An odd root of a negative number is negative, so the cube root of -54 is -3 times the cube root of 2, while an even root of a negative number is not real and uses the imaginary unit i, so the square root of -72 is 6i times the square root of 2. The tool also gives the decimal value, and a list of perfect squares or cubes to look for. Rationalizing a denominator means removing the root from the bottom of a fraction: 1 over the square root of 2 becomes the square root of 2 over 2, and 5 over the square root of 12 becomes 5 times the square root of 3 over 6. Whole numbers up to one trillion are supported.
How do you simplify a square root?▾
Factor the number into primes, take out each pair of equal primes as a single factor in front of the root, and leave any unpaired prime inside. The square root of 72 = 2 x 2 x 2 x 3 x 3 becomes 6 times the square root of 2.
What is simplest radical form?▾
A root is in simplest radical form when the number under it has no perfect square factor (or perfect cube, for a cube root) other than 1, and there is no root in a denominator.
How do you simplify a cube root?▾
Factor into primes and take out each group of three equal primes. The cube root of 54 is 3 times the cube root of 2, because 54 = 2 x 3 x 3 x 3.
What is the square root of a negative number?▾
It is not a real number. It is written with the imaginary unit i, where i squared is -1, so the square root of -72 is 6i times the square root of 2. Odd roots of negative numbers are real and negative.
How do you rationalize a denominator?▾
Multiply the top and bottom of the fraction by the root in the denominator, so the bottom becomes a whole number, then reduce. 1 over the square root of 2 becomes the square root of 2 over 2.
Is my data uploaded?▾
No. The calculation runs in your browser and nothing you enter is sent anywhere.