Multiplying Radicals Calculator
Multiply two coefficient-and-radical factors with equal or different indices, then see the exact simplified form, decimal check, and prime-exponent work.
Radical factors
Live result
All possible perfect sixth-power factors have been extracted.
Simplification steps
- Write the product2 × ³√(12) × 5 × √(18)
- Use a common indexLCM(3, 2) = 6, so the radicands are raised to powers 2 and 3.
- Combine coefficients and radicands10 × ⁶√(839,808)
- Extract perfect sixth powers839,808 = 2⁷ × 3⁸, so one 2 and one 3 leave the radical.
- State the simplified result60 × ⁶√(18)
Prime factor ledger
| Prime | First contribution | Second contribution | Total exponent | Extracted exponent | Remaining exponent |
|---|---|---|---|---|---|
| 2 | 4 | 3 | 7 | 6 | 1 |
| 3 | 2 | 6 | 8 | 6 | 2 |
How to use the multiplying radicals calculator
What this calculator does
This calculator multiplies two expressions in the form a × ⁿ√(b) and c × ᵐ√(d). It finds a common index, combines the prime exponents, extracts every complete group that can move outside the radical, and reduces the remaining index when the exponents permit it. The exact answer is an algebraic identity for the entered integers; the decimal value is only a numerical check. The tool does not multiply sums or binomials, solve equations, or work in the complex-number system.
When to use it
Use it to check homework involving two monomial radical factors, to verify a hand simplification with different root indices, to see why a perfect power leaves a radical, or to compare an exact radical answer with a decimal approximation. The method follows the product property described in OpenStax's lesson on multiplying radical expressions.
How to calculate
- The calculator opens with the complete example 2 × ³√(12) × 5 × √(18), so the result and a validated example workbook are available immediately.
- Replace the three values under First factor, then replace the three values under Second factor. Results update live; there is no separate Calculate button.
- Read Simplified product first. Use Decimal approximation only to check magnitude and sign. Review the five simplification steps and the prime factor ledger when you need to show your work.
- Select Download Excel to export the current inputs, exact result, numeric check, steps, and factor ledger. Reset clears the demonstration and all calculated content; the export becomes unavailable until all six required fields contain a valid set again.
Input guide
First coefficient (a) and Second coefficient (c) are required signed integers from – 1,000,000,000 to 1,000,000,000; examples are 2 and 5. A negative coefficient changes the sign of the product, while zero makes the entire product zero. Enter digits with optional correctly placed thousands separators, not decimals or scientific notation.
First radicand (b) and Second radicand (d) are required whole numbers from 0 to 1,000,000,000; examples are 12 and 18. Larger radicands generally increase the decimal magnitude, but perfect-power factors may leave the radical and enlarge the exact coefficient. Do not enter a negative radicand: this calculator is limited to real nonnegative roots.
First index (n) and Second index (m) are required integers from 2 through 12; examples are 3 and 2. The index tells you which root is taken. A missing small index on a handwritten radical means 2, but the input must still contain 2. Mixed indices are converted through their least common multiple, a standard approach explained in the LibreTexts section on radicals with mixed indices.
Output guide
Simplified product is the exact final identity. Decimal approximation is rounded to four decimal places and is driven by all six inputs. Common index is LCM(n, m). Exact coefficient contains a × c plus every complete common-index power extracted from the combined radical. Residual radicand and Residual index describe the radical that remains; a residual radicand of 1 means the answer is an integer. Combined radicand is the value inside the common-index radical before extraction.
In the Prime factor ledger, Prime identifies a factor; First contribution and Second contribution show the exponents contributed after index conversion; Total exponent adds them; Extracted exponent is the largest multiple of the common index removed; and Remaining exponent stays under the radical. Zero rows are omitted because they carry no factor.
Worked example
For a = 2, b = 12, n = 3, c = 5, d = 18, and m = 2, the common index is LCM(3, 2) = 6. Rewrite ³√(12) as ⁶√(12²) and √(18) as ⁶√(18³). Their product is 10 × ⁶√(839,808). Since 839,808 = 2⁷ × 3⁸, one group of six 2s and one group of six 3s leave the radical. Multiplying those extracted factors by 10 gives 60, while 2¹ × 3² = 18 remains inside. The exact first-open result is therefore 60 × ⁶√(18), approximately 97.1322.
Why the common-index method works
An nth root can be written as a rational exponent: ⁿ√(x) = x^(1/n). To combine roots with different indices, rewrite both fractional exponents with the same denominator. If k = LCM(n, m), then b^(1/n) becomes b^(k/n ÷ k), and d^(1/m) becomes d^(k/m ÷ k). That produces one kth root. OpenStax's overview of radicals and rational exponents develops the same connection.
Common mistakes
- Multiplying radicands directly when the indices differ. Convert to a common index first.
- Extracting a factor after finding only one copy. For a kth root, a prime needs a complete group of k equal factors to leave the radical.
- Rounding the decimal and treating it as the exact answer. Keep the simplified radical form whenever an exact result is required.
- Forgetting to multiply the outside coefficients separately from the radical factors.