Radical Calculator

A radical calculator rewrites a root in simplest radical form by pulling every perfect-power factor out from under the radical sign. The classic case: √72 = 6√2, because 72 splits into 36 × 2 and √36 = 6. The exact form 6√2 carries full precision, while the decimal 8.485281 is only an approximation.

Three neighboring tools cover related questions. The Square Root Calculator is the one to use when you mainly want the decimal value of a root, the Cube Root Calculator focuses on index-3 roots including negative radicands, and the Exponent Calculator handles the reverse direction, raising numbers to whole and fractional powers. Use this page when you need the exact simplified radical, the working that produced it, or an operation on two radicals.

How simplifying a radical works

Every simplification rests on the product rule for radicals, which lets a root be split across a product:

Simplifying means choosing a to be the largest perfect nth power that divides the radicand, so that ⁿ√a comes out as a whole number and only b stays under the sign. In prime-factor language the rule is even simpler: write the radicand as a product of primes, and for every complete group of n identical primes, move one copy outside. A square root takes primes out in pairs, a cube root in threes, a fourth root in fours.

Worked example: simplify √72

Take the square root of 72 and work through it the way the calculator does.

1. Factor the radicand. 72 = 2 × 2 × 2 × 3 × 3.
2. Group in pairs. One pair of 2s and one pair of 3s are complete; a single 2 is left over.
3. Move each pair outside. The pair of 2s becomes a 2 outside, the pair of 3s becomes a 3 outside, and 2 × 3 = 6.
4. Write the answer. √72 = √(36 × 2) = 6√2.
5. Check it. 6√2 = 6 × 1.41421356 ≈ 8.485281, and squaring 8.485281 returns 72 up to rounding. The decimal agrees, so the simplification is right.

The same routine handles bigger numbers. √1,080 = √(36 × 30) = 6√30 ≈ 32.863353, and √2,025 = 45 exactly, because 2,025 is a perfect square.

Worked example: multiplying two radicals

Multiply 3√12 by 2√6. Coefficients multiply with coefficients, radicands with radicands: 3 × 2 = 6 and 12 × 6 = 72, which gives 6√72. Now simplify the new radicand: √72 = 6√2, so the product is 6 × 6√2 = 36√2 ≈ 50.911688. Multiplying the four decimal pieces directly, 3 × 3.464102 × 2 × 2.449490, lands on the same 50.9117. Notice that simplifying after multiplying is usually faster than simplifying each factor first, because a single large radicand is easier to factor once than twice.

Worked example: adding and subtracting like radicals

Radicals behave like algebraic terms: only like radicals combine, meaning the index and the radicand must match after simplifying. Take 5√45 − 2√20. Simplify each term first: √45 = 3√5 so the first term is 15√5, and √20 = 2√5 so the second term is 4√5. Both now carry √5, so subtract the coefficients: 15 − 4 = 11, giving 11√5 ≈ 24.596748. A second case: 3√50 + 2√8 becomes 15√2 + 4√2 = 19√2 ≈ 26.870058. By contrast, √2 + √3 stays exactly as written, because the radicands never match.

Worked example: rationalizing the denominator

Simplest radical form does not allow a radical on the bottom of a fraction. For a one-term denominator, multiply the top and the bottom by that radical. Take 7 ÷ (3√5): multiply both parts by √5, the bottom becomes 3 × 5 = 15, and the result is 7√5 ÷ 15 ≈ 1.043498. For a two-term denominator, multiply by the conjugate, the same two terms with the middle sign flipped, so the difference of squares clears the radical. Take 6 ÷ (4 + √7): multiply by (4 − √7), and the bottom becomes 4² − 7 = 9, giving 6(4 − √7) ÷ 9. Divide every term by 3 and the answer is (8 − 2√7) ÷ 3 ≈ 0.902832, which matches 6 ÷ 6.645751 on a plain calculator.

Square roots 1 to 100 in simplest radical form

This reference table lists every square root from 1 to 100 already simplified. Whole numbers mark the perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100); everything else keeps a prime or prime product under the radical.

Reading the table sideways shows a useful pattern. Radicands that are prime, or a product of distinct primes, never simplify: √2, √15, √35 and √91 are already finished. Radicands built from a square times something small always do: 8, 18, 32, 50, 72 and 98 all reduce to a multiple of √2, and 12, 27, 48, 75 and 300 all reduce to a multiple of √3.

Cube roots and fourth roots, simplified

Higher indexes follow the same rule with bigger groups. A cube root pulls out a factor for every three identical primes, a fourth root for every four. The perfect cubes to watch for are 8, 27, 64, 125, 216, 343, 512 and 1,000; the perfect fourth powers are 16, 81, 256, 625 and 1,296.

An index-3 radicand may also be negative, because a negative number has a real cube root: the cube root of −54 is −3 times the cube root of 2. Even indexes have no real value for a negative radicand, so the calculator flags that case instead of returning a number.

Common denominators to rationalize

These are the fraction forms that show up most often in homework, each shown with the rationalized version and a decimal check.

Two habits pay off here. First, simplify the radical in the denominator before rationalizing, since 4 ÷ (2√3) is easier once you see the 2 cancels. Second, always reduce the final fraction: the conjugate step often leaves a common factor, as in the (8 − 2√7) ÷ 3 example above.

How to use this radical calculator

The calculator updates as you type, so you can try a value and immediately see the exact form, the decimal, and the reasoning. Work through it like this:

1. Pick the operation. Simplify handles one radical, Multiply combines two, Add or subtract merges like radicals, and Rationalize clears a radical out of a denominator.
2. Choose the root index. Leave it at 2 for a square root, or switch to 3, 4 or 5 for a cube, fourth or fifth root.
3. Enter the coefficient and the radicand. The coefficient is the number written in front of the radical sign; the radicand is the number underneath it. Leave the coefficient at 1 if there is none.
4. For a rationalize problem, describe the denominator. Set the whole part to 0 for a one-term bottom such as 3√5, or enter a whole part to trigger the conjugate method for a bottom such as 4 + √7.
5. Read the result. The large figure is the simplest radical form; the tile below shows the decimal to six places, and the step list shows the factorization, the grouping and the reduction.

The quick buttons under the Simplify inputs load the radicands that come up most in class, from √8 to √300, so you can check an answer in a single tap.

Who this calculator is for

• Algebra 1 and Algebra 2 students checking homework where the answer must be given in simplest radical form.
• Geometry students whose Pythagorean-theorem answers land on values such as √117 = 3√13 or the 45-45-90 diagonal √50 = 5√2.
• SAT and ACT test takers, where the multiple-choice options are almost always written as simplified radicals rather than decimals.
• Teachers and tutors who want the intermediate steps on screen while explaining the grouping rule.
• Anyone working with exact values in trigonometry, physics or engineering, where √2 ÷ 2 and √3 ÷ 2 appear constantly and rounding early hurts precision.

Key radical terms explained

• Radical: the whole expression under and including the √ sign.
• Radicand: the number sitting under the radical sign, such as the 72 in √72.
• Index: the small number in the notch of the radical sign. It is 2 for a square root even though it is not written, 3 for a cube root, and so on.
• Coefficient: the number in front of the radical, multiplied by it. In 6√2 the coefficient is 6.
• Like radicals: two radicals with the same index and the same radicand, the only ones that can be added or subtracted.
• Conjugate: the pair a + b√c and a − b√c. Multiplying them gives a² − b²c, a number with no radical, which is why the conjugate clears a two-term denominator.
• Irrational number: a value such as √2 whose decimal never ends and never repeats, so it cannot be written exactly as a fraction.

What changes the result

• The prime factorization of the radicand. This is the whole story for a simplify problem. A radicand rich in repeated primes, like 288 = 2⁵ × 3², reduces a long way (to 12√2); a squarefree radicand like 30 does not reduce at all.
• The index. The same number simplifies differently at different indexes: 32 gives 4√2 as a square root, but 2 times the cube root of 4 as a cube root and exactly 2 as a fifth root.
• Whether you simplify before or after an operation. The answer is identical either way, but the arithmetic is far lighter if you simplify each term before adding and after multiplying.
• The shape of the denominator. One term needs a single multiplication by the radical; two terms need the conjugate and a difference of squares.
• Rounding. The radical form is exact, so any difference you see between the exact and decimal answers is rounding, not a mistake in the algebra.

Tips for simplifying radicals faster

• Test perfect squares from the top down. Try 100, 81, 64, 49, 36, 25, 16, 9 and 4 in that order and stop at the first one that divides evenly; that gets you the fully simplified form in one step.
• Learn the √2 and √3 families. 8, 18, 32, 50, 72, 98, 128 and 200 all give a multiple of √2; 12, 27, 48, 75, 108, 147 and 300 all give a multiple of √3.
• Split off an obvious 4 first. Any even radicand divisible by 4 immediately yields a factor of 2, and repeating this is easier than a full factorization.
• Check with the decimal. Multiply your coefficient by the decimal value of the remaining root and confirm it matches the square root of the original number.
• Keep the radical until the last line. Rounding √2 to 1.41 early can shift a final answer by a percent or more; carry the exact form as long as you can.

Where simplified radicals actually get used

Outside the homework page, radicals show up any time a squared quantity has to be undone. The Pythagorean theorem turns two legs of 6 and 9 into a hypotenuse of √117 = 3√13 ≈ 10.816654, and the diagonal of a 5 by 5 square is √50 = 5√2 ≈ 7.071068. The distance formula in coordinate geometry produces the same kind of answer. In trigonometry, the exact values of the standard angles are radicals: the sine of 45 degrees is √2 ÷ 2 ≈ 0.707107 and the sine of 60 degrees is √3 ÷ 2 ≈ 0.866025, both already rationalized. Quadratic solutions from the discriminant land on radicals too, which is why the Quadratic Formula Calculator reports roots that often need simplifying. Physics uses them for periods and velocities, and statistics uses them for the standard deviation, where a variance is converted back to the original units by taking a square root.

Limitations and assumptions

• The radicand is treated as a whole number. Decimals and fractions under the radical are not simplified symbolically; convert a fraction to a rationalized form by hand, or use the Fraction Calculator first.
• Variables are not supported. Simplify the numeric part here and handle the variable powers separately, taking out one copy for every complete group of the index.
• An even root of a negative number has no real value, so the calculator reports that rather than returning an imaginary result.
• Only the principal root is shown. In an equation such as x² = 72 both 6√2 and −6√2 are solutions, but the radical sign itself refers to the positive one.
• Decimal values are rounded to six places with double-precision arithmetic. The radical form beside them is exact.

How it compares to related calculators

This page answers "what is the simplest radical form, and how do I get there?" If your question is different, another tool fits better:

• For the decimal value of a root or a perfect-square check, use the Square Root Calculator.
• For index-3 roots, including negative radicands, use the Cube Root Calculator.
• To go the other way and raise a number to a power, including fractional exponents, use the Exponent Calculator.
• For quadratic solutions that arrive as radicals, use the Quadratic Formula Calculator.
• To reduce an ordinary fraction to lowest terms, use the Simplify Fractions Calculator.
• For very large or very small values written with powers of ten, use the Scientific Notation Calculator, and for the inverse of exponentiation use the Logarithm Calculator.

Sources

Every result on this page is deterministic mathematics that follows from exact definitions, so no external data source is needed and none is cited:

• The product rule for radicals and the definition of the principal nth root are standard algebraic identities. They hold by definition for non-negative radicands and require no empirical source.
• Prime factorization is unique for every integer above 1 (the fundamental theorem of arithmetic), which is why the simplest radical form of a number is unique.
• The difference of squares, (a + b)(a − b) = a² − b², is the identity behind the conjugate method for rationalizing a two-term denominator.
• All decimal values shown are computed with double-precision floating-point arithmetic as defined by the IEEE 754 standard, and rounded to six places for display.