Simplifying Radicals Calculator — Simplest Radical Form of Square Roots & nth Roots

Simplifying Radicals Calculator

Welcome to Calcugo's simplifying radicals calculator, where we'll take on the most common expressions that include radicals. With the power of magic (i.e., mathematics), we'll rewrite them in an easier form. For instance, the basic case here would be simplifying square roots, that is, values of the form a√b. However, the expressions can be more complicated, so we'll describe how to simplify radicals in general.

Whichever root expression you're dealing with, this simplest radical form calculator will be able to deal with it! If you just want to compute some roots and not necessarily simplify expressions, go to the root calculator instead.

Exponents and roots

Whenever we multiply by the same number several times, we can save ourselves some time (time is money, after all), and instead of repeating the multiplication, write the whole thing using exponents. For example:

5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 = 5⁸

The small number in the superscript tells us how many times we multiply the big number — in this case, we have eight fives. Taking the root (also called radical) is the inverse operation. In other words, while the exponent turns 5 into 5⁸, the (eighth) root makes 5⁸ back into 5:

⁸√(5⁸) = 5

A couple of essential rules govern this simplify radicals calculator:

• The order of a radical must be an integer greater than or equal to 2.
• The number under the root must be a non-negative integer. (Radicals of odd order also apply to negative numbers, and rational numbers have roots too, but for simplicity we limit ourselves to the non-negative integer case.)

How to simplify square roots

Square roots are radicals of order 2 — the inverse operation to squaring a number. For instance, 5² = 25, so √25 = 5. But what is, say, √288? We have 16² = 256 and 17² = 289, so √288 sits somewhere between 16 and 17. Instead of approximating, let's transform it into a prettier form.

The main tool for simplifying radical expressions is prime factorization. Write the number under the radical as a product of primes:

288 = 2 × 2 × 2 × 2 × 2 × 3 × 3 = 2⁵ × 3²

The root is of order 2, so we count how many pairs of identical primes we have: two pairs of 2s, a pair of 3s, and a single 2 left over. We pull the numbers representing each pair in front of the root, and the leftovers stay inside:

√288 = √(2⁵ × 3²) = 2 × 2 × 3 × √2 = 12√2

That is the best expression we can hope for, and it's precisely what the calculator above returns.

How to simplify radical expressions

Prime factorization is still our main tool — we just play around a bit using root and exponent properties. In the calculator you can pick from four options:

1. a × ⁿ√b — For n = 2 this is simplifying square roots. For a different order we look for groups of n identical primes, pull out the number each group represents, and leave the rest inside.
2. a × ⁿ√b + c × ^m√d — We can only add radicals with the same order and same number under the root (n = m and b = d after simplifying). First simplify each term (rule 1), then check whether they can be added.
3. a × ⁿ√b × c × ^m√d — Here a·ⁿ√b × c·^m√d = (a·c)·^k√(b^s × d^t), where k = lcm(n, m), s = k/n and t = k/m. Then apply rule 1.
4. (a × ⁿ√b) / (c × ^m√d) — Similarly, (a·ⁿ√b) / (c·^m√d) = (a / (c·d))·^k√(b^s × d^t) with k = lcm(n, m), s = k/n and t = k·(m−1)/m. The extra m−1 comes from rationalizing the denominator.

One last trick: after we pull out everything we could, the root itself can sometimes be simplified further. If all the primes under the radical have powers sharing a common factor with the root's order, we reduce them the same way we reduce fractions:

^kn√(a^ks × b^kt × …) = ⁿ√(a^s × b^t × …)

For instance:

⁶√(2⁴ × 5¹⁰ × 7²) = 5 × ³√700

Example: using the simplifying radicals calculator

Say you want to simplify √288. Choose the mode Single radical (a · ⁿ√b), leave the coefficient a = 1 and the index n = 2, and type b = 288. The calculator factorizes 288 = 2⁵ × 3², pulls out one 2 and one 3 (and one more 2), and returns the simplest radical form 12√2 — along with the prime-factor table and the step-by-step reasoning.

Want to combine radicals? Switch modes to add 2√3 + 5√3 = 7√3, multiply √2 × ³√2 = ⁶√32, or divide √6 / √2 = √3. Every result is reduced to its simplest radical form automatically.