Enter a radicand (the number under the radical) and an optional index (root degree) to simplify any radical expression to its simplest form. You'll see the simplified radical, the decimal value, and a breakdown of the perfect factor extracted — so √72 becomes 6√2 in one click.
Simplified Radical Form
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Coefficient (Outside the Radical)
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Remaining Radicand (Inside the Radical)
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Decimal Value
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Largest Perfect Factor Extracted
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A radical expression contains a radical symbol (√) with a radicand — the number or expression under the symbol. The small number to the upper-left of the radical is the index, which tells you which root to take (square root, cube root, etc.). When no index is shown, a square root (index = 2) is assumed.
Simplifying a radical means rewriting it so no perfect powers remain under the radical sign. For example, √72 = √(36 × 2) = 6√2, because 36 is a perfect square. The goal is to extract the largest possible perfect factor and move its root outside the radical sign.
The n-th root of x is the same as x raised to the power 1/n: ⁿ√x = x^(1/n). This means √9 = 9^(1/2) = 3, and ∛8 = 8^(1/3) = 2. This relationship lets you apply exponent rules to simplify radical expressions more easily.
Factor the radicand into prime factors and group them by the index. For a square root (index 2), pairs of identical factors come out; for a cube root (index 3), triples come out. Any leftover factors that can't form a complete group stay inside the radical. The product of extracted roots becomes the coefficient outside.
Simplified radicals are easier to compare, add, and subtract — you can only combine like radicals (same index and radicand). Simplification also reduces errors in further calculations and is typically required in algebra, geometry, and standardized test answers.
Yes. Set the index to 3 for cube roots, 4 for fourth roots, and so on up to index 10. The calculator applies the same prime-factorization logic, grouping factors in sets of the given index to find what comes outside the radical.
Radical simplification appears in geometry (e.g., diagonal of a square: d = s√2), physics (wave equations, RMS voltage), engineering (impedance calculations), and statistics (standard deviation formulas). Simplified forms make these expressions far more practical to work with.
The most frequent errors are: extracting a non-perfect factor (e.g., pulling √6 out of √12 as 6√2 instead of 2√3), forgetting to check for larger perfect factors, and confusing the index with the radicand. Always factor fully and verify by squaring (or raising to the n-th power) your answer.