Simplifying Radicals Calculator
Enter the index (2 for a square root, 3 for a cube root, or any whole number), the radicand (the number inside the radical), and an optional outer coefficient. The calculator pulls out all perfect nth-power factors, shows you the prime factorization step by step, and gives you the simplified radical form plus the decimal approximation.
What does simplifying a radical mean?
A radical expression like sqrt(72) is in simplest form when the number inside the radical (the radicand) has no perfect-square factors other than 1. For a cube root the rule is no perfect-cube factors, and for an nth root no perfect nth-power factors. Simplifying pulls those factors out from under the sign and writes them as a coefficient outside. So sqrt(72) becomes 6*sqrt(2) because 72 = 36 x 2 and sqrt(36) = 6. The value of the expression does not change, only the form: 6*sqrt(2) and sqrt(72) are equal to the same decimal, roughly 8.485.
How to simplify a radical step by step
Step 1: write the prime factorization of the radicand. For sqrt(72): 72 = 2^3 x 3^2. Step 2: for each prime, divide its exponent by the root index. The quotient is the number of copies that come out from under the radical, and the remainder stays inside. For sqrt(72) with index 2: the prime 2 has exponent 3 - quotient 1, remainder 1, so one copy of 2 comes out. The prime 3 has exponent 2 - quotient 1, remainder 0, so one copy of 3 comes out and nothing remains inside. Step 3: multiply the factors that came out: 2^1 x 3^1 = 6. The leftover inside is 2^1 = 2. So sqrt(72) = 6*sqrt(2). Step 4: multiply by any outer coefficient. If the original expression was 2*sqrt(72), the answer is 12*sqrt(2).
Cube roots and nth roots
The same prime-factorization method works for any root index. To simplify the cube root of 250: 250 = 2 x 5^3. The prime 5 has exponent 3 - with index 3 the quotient is 1 (one 5 comes out) and the remainder is 0. The prime 2 has exponent 1 - quotient 0, so it stays inside. Result: 5 * cbrt(2). For a fourth root of 162: 162 = 2 x 3^4. The prime 3 has exponent 4, quotient 1 with index 4, so one 3 comes out and nothing remains. Result: 3 * 4th-root(2). The calculator above handles any root index from 2 to 10 automatically.
Radical properties used in simplification
Three fundamental properties make radical simplification work. First, the product rule: nth-root(a x b) = nth-root(a) x nth-root(b). This lets you split the radicand into its perfect-power part and the remainder. Second, nth-root(a^n) = a for positive a: a perfect nth power evaluates to a whole number outside the radical. Third, the coefficient rule: c x nth-root(a) x nth-root(b) = c x nth-root(a x b). Together these justify every step in the method above. These same properties are used to add and subtract radicals (you can only combine like radicals that share the same index and the same simplified radicand) and to multiply or divide radical expressions.
Common perfect squares and their square roots
| Number | Square root | Perfect square? |
|---|---|---|
| 1 | 1 | Yes |
| 2 | 1.414... | No |
| 3 | 1.732... | No |
| 4 | 2 | Yes |
| 5 | 2.236... | No |
| 6 | 2.449... | No |
| 8 | 2.828... | No |
| 9 | 3 | Yes |
| 16 | 4 | Yes |
| 25 | 5 | Yes |
| 36 | 6 | Yes |
| 49 | 7 | Yes |
| 50 | 5*sqrt(2) | No |
| 64 | 8 | Yes |
| 72 | 6*sqrt(2) | No |
| 81 | 9 | Yes |
| 100 | 10 | Yes |
| 144 | 12 | Yes |
| 169 | 13 | Yes |
| 196 | 14 | Yes |
| 225 | 15 | Yes |
Recognizing these instantly makes radical simplification much faster.
Frequently asked questions
How do I simplify sqrt(72)?
Write the prime factorization of 72: 2^3 x 3^2. For each prime, take floor(exponent / 2) copies outside and the remainder inside. For 2: floor(3/2)=1 outside, 1 inside. For 3: floor(2/2)=1 outside, 0 inside. Coefficient: 2^1 x 3^1 = 6. Inside: 2^1 = 2. Answer: 6*sqrt(2), which is approximately 8.485.
What if the radicand is a perfect square (or perfect cube)?
The simplified form is a whole number and no radical sign remains. For example sqrt(144) = 12 (since 144 = 12^2) and cbrt(125) = 5 (since 125 = 5^3). The calculator shows this in the simplified expression field as just the integer.
Can I simplify expressions with a coefficient in front, like 3*sqrt(50)?
Yes. Simplify the radical part first: sqrt(50) = 5*sqrt(2). Then multiply by the outer coefficient: 3 x 5 = 15. Final answer: 15*sqrt(2). Enter 50 as the radicand, 2 as the index, and 3 as the outer coefficient to get this directly.
Why does the decimal value stay the same before and after simplification?
Simplification is purely a change of form, not of value. sqrt(72) and 6*sqrt(2) represent the same real number because 6^2 x 2 = 36 x 2 = 72. The decimal shown in both cases is approximately 8.485281. This equality is a direct consequence of the product rule for radicals: sqrt(a x b) = sqrt(a) x sqrt(b).
What does it mean for a radical to be in simplest form?
A radical nth-root(R) is in simplest form when R has no factor that is a perfect nth power (other than 1). Equivalently, in the prime factorization of R, every prime has an exponent less than n. For sqrt: every prime exponent in the radicand must be 0 or 1. For cbrt: every prime exponent must be 0, 1, or 2.
What is the root index, and what values can it take?
The root index n in nth-root(x) tells you which root you are taking. n=2 is a square root (usually written without the 2), n=3 is a cube root, n=4 is a fourth root, and so on. The index must be a positive integer of 2 or greater. For this calculator you can enter any index from 2 to 10.