If you want to find the radical of any nth degree, then this radical calculator is just for you. The tool on the left is straightforward to use — it only needs two of the three values to estimate the third one. It can also do the reverse calculations!
If you are wondering what radical form is and want to learn about radical notation, read the article below for all the basic information about this math operation. You'll also find the radical definition and some examples explaining the problem of adding and subtracting radicals, together with answers to the questions "How to multiply radicals?" and "How to divide radicals?". It might not be as easy as you think! 😮
If you came here for the simplest root — the one with a degree of 2 — then a dedicated square root calculator gives a more comprehensive description of square roots.
Radical definition — what does radical mean?
In general, the radical is a root of a number x, for which we use the radical symbol n√, with the superscript n describing the degree of the root. The radical definition is:
rn = x
and therefore:
r = x1/n
where r is the result of the radical. In other words, radicalization is the opposite operation to exponentiation, for which the exponent is a fraction. To make writing radicals easier, we use the radical notation:
r = n√x
which means the same thing as the equation above.
Rules for radicals — radical notation
It turns out that you cannot calculate the root of just any number. What are the conditions for radical numbers, then?
- In general, the number you want to root should be positive, although it is possible to find the answer when the degree is odd — e.g., when
n = 3(cube root), since∛8 = 2. - Secondly, the degree of the radical is usually a positive integer. Technically, you can use fractions, but in that case, exponentiation is a better way to present the computation.
To conclude the radical notation, for the radical n√x:
n— should be a positive integer; andx— should be a non-negative real number.
How to use the radical calculator
The radical calculator consists of three values. Choose what you want to find, fill in the remaining two boxes, and we will estimate the third one automatically:
- Number (x) — the number for which you'd like to find the radical. To avoid multiple answers (including complex numbers), we do not allow negative values here, even when the radical degree is odd.
- Radical degree (n) — as mentioned before,
nshould be a positive integer. However, since you can treat a radical as an exponent, it's technically possible to compute the radical for other degrees, so non-integer and negative values are allowed too. - Radical (nth radical, r) — the nth radical is probably the reason you came here 🎉
Use this tool to estimate radicals of any degree, find perfect radicals (by entering a value in the radical field), or just to verify an answer you worked out on paper.
Adding and subtracting radicals
Adding radicals isn't as simple as adding two numbers. For example, this is wrong:
❌ ⁴√7 + ∛4 = ⁷√11 ❌
You can add or subtract radicals only if the degree and the number under the radical sign are the same:
✅ ⁴√7 + 2·⁴√7 = 3·⁴√7 ✅
✅ 5·∛5 + 3·∛5 = 8·∛5 ✅
Sometimes you can manipulate the number under the radical by bringing out a factor that can be rooted. For example, ∛192 = ∛(64 × 3) = ∛64 × ∛3 = 3·∛3, so:
∛192 + ∛3 = 3·∛3 + ∛3 = 4·∛3
Subtracting radicals works the same way — the degree and the number under the radical sign must match:
5·⁶√9 − 2·⁶√9 = 3·⁶√9
How to multiply radicals
This is much easier than adding and subtracting radicals. To multiply two radicals, their degree must be the same; then you multiply the numbers below the radical symbol:
⁴√7 × ⁴√4 = ⁴√(7 × 4) = ⁴√28
If a coefficient sits in front of the radical, simply multiply the coefficients separately:
4·∛3 × 2·∛5 = (4 × 2)·∛(3 × 5) = 8·∛15
You can even multiply radicals of different degrees but the same number under the sign by switching to exponent form and adding the fractional exponents:
⁴√2 × ∛2 = 21/4 × 21/3 = 27/12 = ¹²√(2⁷) = ¹²√128
How to divide radicals
Division follows the same idea as multiplication. For equal degrees, divide the numbers under the radical:
n√a ÷ n√b = n√(a ÷ b)
and coefficients divide separately, just like they multiply.
Radical function — radical graph 📈
Because n√x = x1/n, the radical is a power function with a fractional exponent. For n = 2 you get the familiar square-root curve that grows quickly at first and then flattens out; higher degrees give ever-flatter curves. All radical graphs of non-negative x pass through the points (0, 0) and (1, 1).
FAQs
What is a radical in math?
A radical is the nth root of a number. For rn = x, the radical is r = n√x = x1/n. It is the inverse operation of raising a number to the nth power.
How do I find the nth root of a number?
Raise the number to the power of 1/n: n√x = x1/n. For example, the cube root of 64 is 641/3 = 4, because 4³ = 64.
What is a perfect radical?
A radical is "perfect" when its result is a whole number — that is, when x is a perfect power of an integer for the chosen degree. For instance, ∛27 = 3 and ⁴√81 = 3 are perfect radicals.
Can the number under the radical be negative?
Only for odd degrees does a real answer exist (e.g., ∛(−8) = −2). For even degrees, a negative radicand has no real root. To keep results unambiguous, this calculator accepts only non-negative numbers under the radical.