Radical Calculator: Simplify and Evaluate Roots

A radical calculator evaluates roots and helps simplify expressions such as √72 or ∛(-27). Enter the number under the radical and the root’s index; the calculator returns the root’s value, with a separate exact form when the result can be simplified neatly.

Formula

The value of the kth root of n is n^1/k. In symbols, ⁿ√n = n^1/k, where n is the radicand (the number inside the radical) and k is the root index. Both are dimensionless numbers here. For a square root, k = 2; for a cube root, k = 3.

The calculator’s first output is the real value n^1/k. Its second output uses √n, which represents the simplified square-root expression when k = 2. For indices other than 2, that second output is not a general radical-simplification result; use the first output for the numerical root and apply the factorization method below to simplify the expression exactly.

Worked Example

Using the default inputs, n = 72 and k = 2:

1. Apply the root formula: 72^1/2 = √72.
2. Factor 72 to find a perfect square: 72 = 36 × 2.
3. Take the square root of the perfect-square factor: √72 = √36 × √2 = 6√2.
4. Evaluate the decimal value: 6√2 ≈ 8.4853.

The calculator rounds the first output to four decimal places, giving 8.4853. The second output evaluates √72 numerically as well; it does not display the symbolic expression 6√2.

What a Radical Calculator Does

A radical is a symbol that indicates a root. The square-root symbol √ is the most familiar, but roots can have other indices. The index is written at the upper left of the radical: ³√8 means the cube root of 8, or the number that, when multiplied by itself three times, equals 8. When no index is shown, the index is 2.

A calculator is useful for checking a numerical result, comparing roots, or evaluating roots that do not simplify to tidy integers. For example, √2 is irrational, so its decimal expansion continues without ending or repeating. A calculator can show an approximation, but the exact value remains √2.

• √49 = 7, because 7² = 49.
• ³√125 = 5, because 5³ = 125.
• ⁴√16 = 2, because 2⁴ = 16.
• √10 ≈ 3.1623; this is an approximation, not an exact replacement for √10.

How to Simplify a Radical

To simplify a square root, factor the radicand into a perfect square multiplied by any remaining factor. Take the square root of the perfect-square part and leave the rest inside the radical. For √72, the factorization 72 = 36 × 2 gives √72 = 6√2.

For a root with index k, look for factors that are perfect kth powers. For instance, ³√54 = ³√(27 × 2) = 3³√2. This method preserves the exact value, unlike rounding a decimal. A result is in simplest radical form when no factor left inside the radical is a perfect power matching its index.

Some calculators show only a decimal approximation. If you need an exact expression for algebra, keep the radical form or simplify it by factoring; do not treat a rounded decimal as exact.

Choosing the Inputs and Reading the Answer

Enter the radicand as n and the root index as k. The index must be a positive whole number. The example calculator’s numerical formula uses real-number exponentiation, so it is suitable for nonnegative radicands. For negative inputs, the real-valued rules depend on whether the index is odd or even: an odd root of a negative number is real, while an even root is not real.

• For √(-9), there is no real-number answer; in complex numbers, the principal square root is 3i.
• For ³√(-8), the real answer is -2.
• For a positive radicand and an even index, the principal root is the nonnegative root. The equation x² = 9 has two solutions, 3 and -3, but √9 means 3.

That last distinction is a frequent source of confusion: a radical symbol names the principal root, while solving an equation may produce multiple roots. Also avoid rounding intermediate steps too early. Keep the exact radical or extra decimal places until the final result, especially when using it in a later calculation.

Limits and Common Mistakes

Radicals are not distributed over addition or subtraction: √(a + b) generally does not equal √a + √b. For example, √(9 + 16) = 5, whereas √9 + √16 = 7. The product rule √(ab) = √a × √b is valid for nonnegative real a and b, but the conditions matter when negative or complex values are involved.

Likewise, √(x²) is |x| for real x, not always x. If x = -4, then √(x²) = √16 = 4. Keep track of parentheses and the scope of the radical bar: √(x + 1) is different from √x + 1.

A numerical calculator may also have limits on the size or precision of values it can represent. A displayed decimal is rounded to the stated number of places; it does not imply that the exact answer terminates there. For exact work, use simplified radical notation when possible.

Frequently Asked Questions

How do I calculate a square root?
Use the square-root function or enter the radicand with index 2. For example, √72 ≈ 8.4853, and its exact simplified form is 6√2.

How do I simplify a radical?
Factor the radicand and pull out any perfect power matching the root’s index. For √72, take out 36 because 72 = 36 × 2; the result is 6√2.

What is the difference between a square root and a cube root?
A square root is a number that squares to the radicand; a cube root is a number that cubes to it. For example, √25 = 5 and ³√125 = 5.

Can you take the square root of a negative number?
Not as a real number. In the complex numbers, √(-9) is 3i, where i² = -1. Negative numbers do have real odd roots, such as ³√(-8) = -2.

Is a calculator’s radical answer exact?
A decimal display is normally rounded, so it is an approximation. An exact answer may be an integer or a simplified radical, such as 6√2 for √72.

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in n | Number under radical | - | 72 | 0
in k | Root index | - | 2 | 1
out root | Radical value | - | n ^ (1 / k) | 4
out simplified | Simplified value | - | sqrt(n) | 4