Simplify Radicals Calculator

What Is Simplified Radical Form?

A square root √n is in simplified radical form when the radicand (the number under the radical) has no perfect square factors other than 1. In other words, √n = a√b where b is square-free — none of its prime factors appears more than once.

For example: √72 is not simplified because 72 = 36 · 2 = 6² · 2, and 36 is a perfect square. The simplified form is 6√2. On the other hand, √30 = √(2·3·5) is already simplified — no prime factor is repeated.

Simplified radical form is the standard mathematical convention for expressing irrational numbers. It is exact (unlike decimal approximations), concise, and makes algebraic manipulation much easier. When two students get "3√5" and "√45" for the same problem, simplified form reveals they are the same: √45 = √(9·5) = 3√5.

How to Simplify a Square Root: The Prime Factorization Method

1. Factor the radicand completely into prime factors. Example: 180 = 2² · 3² · 5.
2. Group prime factors in pairs. Each pair of identical primes contributes one factor outside the radical: √(p²) = p.
3. Bring paired factors outside the radical. Any remaining unpaired factors stay inside.
4. Multiply the outside factors to get the coefficient a, and the inside factors to get the new radicand b.

For 180: 180 = 2²·3²·5. Pairs: one pair of 2s and one pair of 3s. Outside: 2·3 = 6. Inside: 5. Result: 6√5.

Worked Example 1: √72

Check: (6√2)² = 36 · 2 = 72 ✓. The radicand 2 has no perfect square factor, so 6√2 is fully simplified.

Worked Example 2: √200

Check: (10√2)² = 100 · 2 = 200 ✓.

Worked Example 3: Perfect Square (√144)

When the radicand is a perfect square, the result is a whole number. The calculator detects this and reports "√144 = 12" without a residual radical.

Worked Example 4: Already Simplified (√30)

When no prime factor appears twice, the radical is already in simplified form. The coefficient is 1 and the radicand is unchanged.

Simplified Radical Form vs. Decimal Approximation

Decimal approximations like √2 ≈ 1.41421356 are rounded — they are never exact. Simplified radical form 1·√2 is exact, regardless of how many decimal places you write.

In algebra courses, "exact form" is required. Writing √72 ≈ 8.485 loses precision; writing 6√2 is exact. This matters in geometry (exact area of a square with diagonal 12 is 72, and √72 = 6√2) and in proofs where exact values are essential.

Simplified radical form also makes arithmetic easier: 3√5 + 2√5 = 5√5 (add coefficients, like like terms). This would be opaque in decimal form (6.708... + 4.472... = 11.180..., and recognizing this as 5√5 requires working backwards).

Radicals in Linear Algebra: Where This Comes Up

Simplified radical form appears constantly in linear algebra because vector lengths (norms) involve square roots. The magnitude of a vector (a, b) is √(a² + b²). Normalizing a vector requires dividing by this length — and keeping the exact form avoids rounding errors.

Eigenvalues of matrices often involve radicals. For a 2×2 matrix with characteristic polynomial λ² − 4λ + 1 = 0, the eigenvalues are λ = 2 ± √3. Keeping these in radical form is essential for symbolic computation.

The Gram-Schmidt process generates orthonormal bases by normalizing vectors. Each normalization step involves a square root — keeping them in simplified form prevents error accumulation.

Rationalizing the Denominator

When a radical appears in a denominator, the standard form requires removing it. The technique depends on what's in the denominator:

Simple radical denominator: multiply numerator and denominator by the same radical.

Binomial with radical: multiply by the conjugate (change the sign between terms).

The key: (a + √b)(a − √b) = a² − b eliminates the radical. Rationalizing is required in standard simplified form and essential in calculus when evaluating limits involving radicals.

Adding, Subtracting, and Multiplying Radicals

Adding and subtracting: only radicals with the same radicand can be combined directly. Always simplify each radical first, then check if radicands match.

Multiplying radicals: √a · √b = √(ab), then simplify the product.

This Simplified Radical Form Calculator in Linear Algebra

This radical form calculator and simplified radical form calculator appears constantly in linear algebra because vector norms involve square roots. The Euclidean norm of v = ⟨1, 2, 3⟩ is |v| = √(1+4+9) = √14. Since 14 = 2·7 has no repeated prime factor, √14 is already in simplified radical form.

Eigenvalues frequently involve radicals. For a matrix with characteristic polynomial λ² − 6λ + 7 = 0, the eigenvalues are λ = (6 ± √(36−28))/2 = (6 ± √8)/2 = (6 ± 2√2)/2 = 3 ± √2. Keeping eigenvalues in simplified radical form is essential for exact symbolic computation.

The Gram-Schmidt process normalizes each vector by dividing by its norm. If the norm is 3√5, normalizing requires dividing by 3√5 — and rationalizing gives clean form. Use this simplified radical form calculator to simplify norms encountered during Gram-Schmidt, eigenvalue computation, or any other linear algebra procedure requiring exact square roots.

For related calculations, see our dot product calculator (for computing the norms v·v = |v|² before taking the square root), the RREF calculator for the matrix operations that generate these radical expressions, and the matrix inverse calculator when eigenvalue computations involve radicals in the characteristic polynomial.

Radicals in Matrix Norm Calculations

When you compute the Frobenius norm of a matrix (the square root of the sum of squared entries), you get a radical expression. For example, the Frobenius norm of [[1,2],[3,4]] is √(1+4+9+16) = √30. Since 30 = 2·3·5 has no repeated prime, √30 is already simplified. Use this calculator to simplify any such norm before continuing with further matrix operations.

Radicals also appear in eigenvalue calculations. If you compute eigenvalues using the determinant calculator to evaluate det(A − λI) = 0 and solve a quadratic, the discriminant involves a square root. For a full introduction to why eigenvalues matter in linear algebra, see our linear algebra basics guide.

Cube Roots and nth Roots

The same prime factorization approach extends to cube roots (∛n) and nth roots: group prime factors in triples (or n-tuples) instead of pairs.

For the nth root ⁿ√k: factor k into primes, group prime factors in sets of n. Each complete group of n identical primes contributes one factor outside the radical; remaining primes stay inside.

This calculator simplifies square roots, the most common case in pre-calculus and linear algebra. For cube roots and higher, apply the same prime factorization method manually using the same grouping principle.

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