Simplify Radicals Calculator

Factor the number under the root, find the largest perfect-power factor that matches the root's index and take its root outside. For √72, the largest perfect-square factor is 36, so √72 = √36 x √2 = 6√2. The calculator uses prime factorization, so it never misses a factor.

Look for perfect squares among the factors of the radicand: 4, 9, 16, 25, 36 and so on. Write the radicand as a perfect square times what is left, split with the product rule and take the square root of the perfect square. √50 = √25 x √2 = 5√2. The square root table above lists the results from 2 to 100.

A radical is in simplest form when the number under it has no perfect-power factor that matches the index, there are no fractions under the root and no radical is left in a denominator. 6√2 is simplest, while √72 and 2√18 are not. The calculator returns this form.

Find the largest perfect cube factor: 8, 27, 64, 125 and so on. ∛54 = ∛27 x ∛2 = 3∛2. If the radicand has no cube factor, such as ∛10, it is already simplified. Set the index to 3 in the calculator to do any cube root.

Split each variable's exponent by the index. For square roots, x⁵ has two pairs and one left, so √(x⁵) = x²√x. Write x^m outside as x^{m ÷ index, rounded down} and keep the remainder inside. The calculator handles one variable with a positive-variable assumption.

Simplify the radical as usual, then multiply the number taken outside by the coefficient already in front. 4√48: √48 = 4√3, so the result is 4 × 4√3 = 16√3. Enter the coefficient in the calculator and it does the multiplication.

72 = 36 × 2 and 36 is a perfect square, so √72 = √36 x √2 = 6√2, which is about 8.485. Using primes, 72 = 2³ × 3²: the 3² and one pair of 2s come out, leaving one 2 inside.

48 = 16 × 3 and 16 is a perfect square, so √48 = 4√3, about 6.928. A common mistake is stopping at 2√12, which still has the perfect square 4 inside, so it needs one more step to reach the simplest form 4√3. Always check the remaining radicand again.

50 = 25 × 2, so √50 = √25 x √2 = 5√2, which is about 7.071. The calculator gives this result and the steps for any radicand. If the radical has a coefficient of 2 in front, the answer becomes 10√2.

32 = 16 × 2, so √32 = 4√2, about 5.657. The same answer comes from primes: 32 = 2⁵, which gives two pairs of 2s outside and one 2 left inside, so 2 × 2 = 4 outside and √2 inside.

18 = 9 × 2, so √18 = 3√2, which is about 4.243. Since 2 has no square factor other than 1, the radical 3√2 cannot be simplified any further. You can confirm by squaring: 3√2 squared is 9 × 2 = 18, which matches √18 squared.

200 = 100 × 2, so √200 = 10√2, about 14.142. Finding the largest perfect square quickly saves work, but if you start with 4 × 50 you simply need a second step: 2√50 = 2 × 5√2 = 10√2.

A negative number under an even root is not real, so write i for the square root of -1 and simplify the rest. √-12 = i√12 = 2i√3. The calculator displays this imaginary form and gives no decimal value because the result is not a real number.

Odd roots of negatives are real, so the sign moves outside. ∛(-54) = -∛54 = -3∛2. The calculator handles odd indices with negative radicands this way and does not introduce an imaginary i, which appears only for even roots of negative numbers.

Look for a perfect fourth-power factor: 16, 81, 256 and so on. ∜64 = ∜(16 × 4) = 2∜4. Since ∜4 equals √2, some courses reduce the index and write 2√2, the same number. The calculator removes the fourth-power factor and shows the decimal value.

Use prime factorization. For an index n, each prime factor p^e contributes p^{e ÷ n, rounded down} to the outside and p^{e mod n} stays inside. This works for any index, which is why the calculator accepts indices from 2 to 12.

The product rule says that the nth root of a product is the product of the nth roots: root(ab) = root(a) x root(b), for non-negative numbers. It is what lets you pull a perfect square out of a radical, as in √72 = √36 x √2.

The quotient rule says the nth root of a fraction is the root of the numerator over the root of the denominator: root(a/b) = root(a) / root(b). So √(9/4) = 3/2 and √(50/9) = 5√2 / 3. It holds when the denominator is not zero and both parts are non-negative.

Simplify each radical first, then combine the ones that have the same radicand and index, just as you combine like terms. √12 + √27 = 2√3 + 3√3 = 5√3. Radicals such as √2 + √3 are unlike and cannot be combined.

Multiply the coefficients and the radicands separately, then simplify. 2√3 × 4√6 = 8√18 = 8 × 3√2 = 24√2. This uses the product rule in the opposite direction. You can enter the product's radicand in the calculator to finish the simplification.

A prime has no factors other than 1 and itself, so there is no perfect square factor to take out. √7, √11 and √13 are already in simplest form. The calculator confirms this by showing the radicand unchanged and giving only the decimal value.

Square the simplified form and compare it with the original radicand. 6√2 squared is 36 × 2 = 72, which matches. You can also compare decimals: 6 × 1.41421 = 8.48528, and √72 is 8.48528 as well. The calculator shows the decimal value so you can check.

A perfect square is the square of a whole number: 1, 4, 9, 16, 25, 36 and so on. Its square root is a whole number. Knowing the squares up to 15 or so makes simplifying much faster, and the table above lists them to 30 squared.

A perfect cube is a whole number cubed: 1, 8, 27, 64, 125, 216 and so on. Cube roots simplify by pulling out these factors, so ∛54 = ∛27 x ∛2 = 3∛2. The table above lists cubes, fourth powers and fifth powers up to 10.

Multiply the top and bottom by the radical in the denominator so the denominator becomes a whole number. 1/√2 = √2/2. This calculator simplifies one radical and does not rationalize fractions, but simplifying the radical first is the usual first step.

They name the same thing from two angles. The radical is the symbol and the expression under it; the root is the value it stands for. In √72, the radicand is 72, the index is 2, and the root is about 8.485. The simplified radical 6√2 is an exact form of that same root.

No. A radical is already in simplest form when the radicand has no perfect-power factor matching the index. √10 and ∛12 are examples of this. Entering them in the calculator returns the radical unchanged, with the decimal value shown beside it.

The radicand can be up to 1,000,000,000,000 (one trillion), using exact integer arithmetic so there is no rounding. For example, 999,999,999,999 simplifies to 3√111111111111. Larger numbers could take too long to factor inside a web browser, so the calculator stops there.

12 = 4 × 3 and 4 is a perfect square, so √12 = √4 x √3 = 2√3, which is about 3.464. With prime factors, 12 = 2² × 3, so the pair of 2s comes out as a single 2 and the 3 stays inside.

75 = 25 × 3 and 25 is a perfect square, so √75 = √25 x √3 = 5√3, which is about 8.660. The prime factors are 3 × 5², so the pair of 5s leaves and the single 3 stays under the root.

98 = 49 × 2 and 49 is a perfect square, so √98 = 7√2, about 9.899. The prime factors are 2 × 7², so the pair of 7s leaves the radical as a single 7 and the 2 remains inside.

108 = 36 × 3 and 36 is a perfect square, so √108 = 6√3, about 10.392. The prime factors are 2² × 3³: one 2 and one 3 come out, giving 6, and one 3 stays inside. Check by squaring: 36 × 3 = 108.

16 = 8 × 2 and 8 is a perfect cube, so ∛16 = ∛8 x ∛2 = 2∛2, about 2.520. The prime factors are 2⁴: one group of three 2s comes out as a 2, and a single 2 stays inside.

The calculator takes a whole-number coefficient and radicand, so simplify the radical first, then handle the fraction by hand. For example, for (1/2)√72, enter √72 to get 6√2 and multiply the coefficient: (1/2) × 6√2 = 3√2. The same approach works for any fractional coefficient.