Simplify Radicals Calculator

What does it mean to simplify radicals?

To simplify a radical means rewriting ⁿ√x as a·ⁿ√b where b has no perfect n-th-power factors left. For example, √72 becomes 6√2 because 72 = 2³ × 3², and the squared factors 2² × 3² come out as 2 × 3 = 6.

How do you simplify √72 step by step?

Factor 72 into primes: 72 = 2³ × 3². Group the factors in pairs: 2³ = 2² × 2, and 3² is already a pair. Each pair comes out of the square root: 2 × 3 = 6, leaving √2. So √72 = 6√2. Check: (6√2)² = 36 × 2 = 72.

What is the largest perfect square factor method?

Find the largest perfect square dividing the radicand, take its square root outside, and leave the rest inside. For √50, the largest perfect square factor is 25, so √50 = √(25 × 2) = 5√2. This calculator does the equivalent prime-factorization method automatically.

Can you simplify cube roots like ∛54?

Yes. Group prime factors in threes instead of pairs. 54 = 2 × 3³, so 3³ comes out as 3 and ∛54 = 3∛2. Check: (3∛2)³ = 27 × 2 = 54. Odd roots also work for negative radicands: ∛(−54) = −3∛2.

How do you add radicals? What is √18 + √8?

You can only add or subtract like radicals — same root index and same radicand. Simplify each term first: √18 = 3√2 and √8 = 2√2. Then add the coefficients: 3 + 2 = 5, so √18 + √8 = 5√2.

Why can't you combine √2 + √3?

Because 2 and 3 are different radicands, they are unlike radicals — just as you cannot combine 2x + 3y into one term in algebra. √2 + √3 is already in simplest form.

What is the difference between simplifying a radical and finding its decimal value?

Simplifying keeps the answer exact and symbolic: √72 = 6√2. Evaluating gives an approximate decimal: √72 ≈ 8.48528. This tool simplifies symbolically; for decimal values of any ⁿ√x use our Nth Root Calculator.

What are the limits of this calculator?

The radicand x must be an integer with |x| ≤ 10¹² and the root index n an integer from 2 to 10. Even roots of negative numbers are not real, so √(−4) is rejected; odd roots of negatives are fine.