A radical is a mathematical expression that involves a root — most commonly a square root (√), cube root (∛), or nth root. The symbol √ is called the radical sign; the number under it is the radicand; and the small number written to the upper-left (when not 2) is the index, which specifies the degree of the root. For example, in ∛27, the index is 3, the radicand is 27, and the result is 3 because 3³ = 27.

How do you simplify a square root?

To simplify √n, find the prime factorization of n, then extract pairs of prime factors (since √(a²) = a). For example, √72: 72 = 2² × 2 × 3² = 4 × 18. Prime factorization: 72 = 2³ × 3². Extract pairs: 2² gives one 2 outside, 3² gives one 3 outside. Result: 2 × 3 × √2 = 6√2. The leftover unpaired factor (2) stays under the radical. Always verify: (6√2)² = 36 × 2 = 72. ✓

What does it mean to simplify a radical?

A radical is fully simplified when: (1) the radicand has no perfect square factors (for square roots), no perfect cube factors (for cube roots), and so on; (2) the radicand is not a fraction; and (3) no radicals appear in the denominator of a fraction. Simplifying a radical finds the exact form with the smallest possible radicand — for example, 6√2 instead of √72. This makes arithmetic easier and helps identify when two expressions are equal.

How do you simplify √72?

√72 = √(36 × 2) = √36 × √2 = 6√2. The key is identifying 36 as the largest perfect-square factor of 72. You can also use prime factorization: 72 = 2³ × 3² = (2¹ × 3¹)² × 2¹, so the coefficient outside is 2 × 3 = 6 and the remaining radicand is 2. The decimal approximation is 6√2 ≈ 6 × 1.41421 ≈ 8.4853.

The nth root of a number x (written ⁿ√x) is the value that, when raised to the power n, gives x. The square root (n = 2) and cube root (n = 3) are the most common. The fourth root (n = 4) of 81 is 3, because 3⁴ = 81. For any positive number, even-index roots always have a positive real result; odd-index roots can handle negative numbers (∛−8 = −2 because (−2)³ = −8). Nth roots can also be written using rational exponents: ⁿ√x = x^(1/n).

How do you handle negative radicands?

Whether a negative radicand has a real result depends on the index. Odd indices (cube root, 5th root, etc.) always work with negative numbers: ∛−8 = −2, ∛−27 = −3. Even indices (square root, 4th root, etc.) cannot produce real results from negative radicands — √−4 is not a real number (it equals 2i in the complex number system). If you encounter √−4 on a real-number problem, check whether you made a sign error in your inputs.

Where do square roots and radicals appear in real life?

Square roots appear constantly in practical math: the Pythagorean theorem (c = √(a² + b²)) gives distances in navigation, construction, and GPS. The quadratic formula contains √(b² − 4ac). Standard deviation is the square root of variance. In physics, the speed of sound, RMS voltage, and many wave equations involve square roots. In finance, volatility (risk) is measured as the square root of variance over time. Wherever you have a "square area and want the side length," you need a square root.

What does rationalizing the denominator mean?

Rationalizing the denominator means rewriting a fraction so the denominator contains no radical. For example, 1/√2 is not rationalized. Multiply numerator and denominator by √2: (1 × √2) / (√2 × √2) = √2 / 2. Now the denominator is a whole number. For fractions with surds like 1/(3 + √5), multiply by the conjugate (3 − √5): the denominator becomes (3 + √5)(3 − √5) = 9 − 5 = 4. Rationalization is standard in algebra because irrational denominators are harder to compare and add.

Radical Calculator — Simplify & Evaluate Square Roots

Simplifies square roots and nth roots to their exact form (e.g. 6√2) and evaluates the decimal approximation — shows prime factorization steps.

How to Use the Radical Calculator

This radical calculator simplifies square roots and nth roots to their exact form — enter the radicand (the number under the radical sign) and the index (the root degree — 2 for square root, 3 for cube root, etc.). Use the quick-select buttons for the most common roots. The calculator immediately returns the simplified exact form (like 6√2 for √72), the decimal approximation, the result type (rational or irrational), and the prime factorization used to perform the simplification.

Simplified radicals often appear in geometry — for example, the diagonal of a unit square is √2 ≈ 1.414. The Pythagorean theorem calculator frequently produces radical expressions when the triangle sides are not a Pythagorean triple.

How to Simplify a Radical

Simplifying a radical means rewriting it so the radicand has no perfect-power factors. The process uses prime factorization:

Find the prime factorization of the radicand. Example: 72 = 2³ × 3².
Group prime factors by the index. For √ (index 2), group in pairs. For ∛ (index 3), group in triples.
Extract complete groups outside the radical. One complete group of 2 comes out as one factor. Two complete groups come out as a squared factor, and so on.
Multiply remaining factors left inside the radical.

Example — √72: 72 = 2³ × 3². The 3² is a complete pair → 3 comes outside. The 2³ = 2² × 2¹ — the 2² is a complete pair → 2 comes outside; the leftover 2¹ stays inside. Result: 2 × 3 × √2 = 6√2.

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Square Root Simplification Examples

Here are common square root simplifications to recognize on sight:

√4 = 2 (perfect square)
√8 = 2√2
√12 = 2√3
√18 = 3√2
√20 = 2√5
√24 = 2√6
√27 = 3√3
√32 = 4√2
√45 = 3√5
√50 = 5√2
√72 = 6√2
√75 = 5√3
√98 = 7√2
√200 = 10√2

When the radicand is a perfect square (4, 9, 16, 25, 36, 49, 64, 81, 100, ...), the result is a whole number with no radical remaining.

Cube Roots and Higher-Index Radicals

The same prime-factorization method applies to any index. For cube roots (∛), extract groups of three identical prime factors:

∛8 = ∛(2³) = 2
∛27 = ∛(3³) = 3
∛54 = ∛(2 × 3³) = 3∛2 (since 3³ comes out, 2 stays inside)
∛72 = ∛(2³ × 3²) = 2∛9 (since 2³ comes out, 3² = 9 stays inside)
∛−64 = −4 (negative cube roots are real: (−4)³ = −64)

For even-index roots of negative numbers (like √−4), no real number solution exists. These require complex numbers, which are outside the scope of this calculator.

Rational vs. Irrational Results

A radical result is rational when the radicand is a perfect power for the given index — the result simplifies to an integer or fraction. Examples: √9 = 3 (rational), ∛−125 = −5 (rational), ⁴√16 = 2 (rational).

A radical result is irrational when the radicand is not a perfect power — the decimal never terminates or repeats. Examples: √2 ≈ 1.41421356..., √3 ≈ 1.73205080..., ∛2 ≈ 1.25992105.... These are important constants in geometry and physics. Use our rounding calculator to round irrational decimal approximations to a desired number of places.

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Nth Roots and Rational Exponents

Every radical can be written using a rational (fractional) exponent: ⁿ√x = x^(1/n). This equivalence is important in algebra because it allows radical expressions to be simplified using the standard rules of exponents. For example:

√x = x^(1/2)
∛x = x^(1/3)
∜x = x^(1/4)
⁵√x³ = x^(3/5) — a radical with a coefficient in the radicand

The exponent rules (product rule, power rule, quotient rule) all apply to rational exponents, making them a powerful tool for simplifying complex radical expressions. For converting results between decimal and fraction form, see our decimal to fraction calculator.

Sources & References

Simplifying Radicals — Math Is Fun — Math Is Fun
Radicals and Rational Exponents — Khan Academy — Khan Academy
nth Root — Britannica — Britannica

Radical Calculator