Simplifying Radicals Calculator
Reduce one radical or combine two radicals in exact form, with prime-factor steps and a decimal check.
Expression inputs
Live result
Simplified expression
6∛2
Original expression
3∛16
Decimal approximation
≈ 7.5595262994
Prime factorization
16 = 2⁴
Exact-form status
Fully simplified
Simplification steps
| Step | Transformation | Reason |
|---|---|---|
| 1 | 3∛16 | Write the original radical expression. |
| 2 | 16 = 2⁴ | Prime-factor the first radicand. |
| 3 | 6∛2 | Move every complete group of three equal prime factors outside the root. |
The table is generated from the same exact model used by the result cards and Excel workbook.
How to use the simplifying radicals calculator
What this calculator does
This calculator rewrites radical expressions in simplest exact form. It can simplify one term of the form a × ⁿ√b, add two radical terms, multiply them, or divide one by the other. The model factors each non-negative integer radicand into primes, removes complete groups that match the root index, combines compatible terms, and rationalizes a radical denominator when needed. It also shows a decimal approximation so you can check magnitude, but the exact radical expression is the primary result. The tool works with integer coefficients and integer radicands; it is not a symbolic algebra system for variables, complex roots, or arbitrary fractions inside a radicand.
Use it to check homework, verify hand calculations, compare equivalent exact forms, or inspect why two radicals can or cannot be combined. The underlying product, quotient, and like-radical rules are summarized in the OpenStax guide to radicals and rational exponents.
How to calculate
- The calculator opens with a complete demonstration: 3∛16. Its exact result, 6∛2, is already calculated, and the Excel workbook is immediately available.
- Choose an option under Expression. Selecting Sum, Product, or Quotient reveals the second radical inputs.
- Replace the coefficients, radicands, and root indices. Results update automatically after every valid edit; there is no separate Calculate button.
- Read Simplified expression first, then use Decimal approximation, Prime factorization, and the step table to audit the transformation.
- Select Download Excel to export the current validated inputs, outputs, and steps as a real .xlsx workbook. Reset clears the demonstration values and results; the export button remains disabled until a complete valid expression is entered again.
Input guide
Expression is required and selects Single radical, Sum, Product, or Quotient. The choice changes the operation, not the meaning of the fields. For example, choose Product to simplify 2√6 × ⁴√64. A common mistake is leaving Quotient selected while intending multiplication.
Coefficient a is the signed whole-number multiplier on the first radical. It accepts integers from – 1,000,000 through 1,000,000; a blank coefficient means 1. A value of 3 in 3∛16 triples the root value. A zero coefficient makes the entire first term zero. Do not enter decimals, fractions, commas, or scientific notation.
Radicand b is the required non-negative integer under the first radical, from 0 through 1,000,000,000. In 3∛16, b is 16. Larger radicands do not necessarily create more complicated answers: a perfect nth power can simplify completely. Negative radicands are deliberately rejected even when an odd root would be real, matching the calculator's non-negative-integer scope.
Root index n is the required whole number from 2 through 12. Use 2 for a square root, 3 for a cube root, and so on. Increasing the index changes which prime factors can leave the radical. Do not enter 1, 0, a decimal, or a negative index.
Coefficient c, Radicand d, and Root index m describe the second term and follow the same formats and limits. They are required only for Sum, Product, and Quotient. In a quotient, c and d must produce a nonzero denominator; c = 0 or d = 0 is invalid. Blank c means 1.
Output guide
Simplified expression is the exact algebraic result. It may be an integer, a rational coefficient times a radical, a sum of unlike radicals, or zero. Original expression restates the current inputs so you can detect an entry mistake. Decimal approximation evaluates the original expression numerically and is an estimate, not a replacement for the exact form. Prime factorization lists the prime powers used to identify complete groups. Exact-form status confirms whether the result is fully reduced or, for a sum, whether unlike radicals must remain separate.
The Simplification steps table records each transformation. Its Step column gives the sequence number, Transformation shows the expression at that stage, and Reason explains the governing rule. The summary pills repeat the selected operation, the active root type or pair of indices, and the exact-status classification.
Worked example
The startup example uses a = 3, b = 16, and n = 3. Prime factorization gives 16 = 2⁴ = 2³ × 2. One full group of three 2s leaves the cube root as a factor of 2, while one 2 remains under the radical:
The exact first-open result is therefore 6∛2. The decimal is only a check. Prime factorization is reliable because every positive integer has a unique prime decomposition; see the OpenStax prime factorization and LCM lesson for the procedure used by the calculator.
How radical simplification works
For an nth root, every complete group of n identical prime factors can move outside the radical as one factor. If b = 2⁷ × 3² and n = 3, then two complete groups of three 2s leave the radical, while one 2 and both 3s remain inside. The same logic applies to square, cube, fourth, and higher roots.
Two radical terms add only after both have been simplified and only when they have the same root index and the same remaining radicand. For instance, √8 + √18 becomes 2√2 + 3√2 = 5√2. By contrast, √2 + √3 remains a sum because the radicals are unlike.
For products and quotients with different indices, the calculator uses the least common multiple of the indices to express both radicals with one compatible root order. It then adds prime exponents for multiplication or subtracts them for division. In a quotient, negative prime exponents are shifted into the rational coefficient so no radical remains in the denominator. The concept of least common multiple is covered in the same prime factorization and LCM reference.