Multiplying Radicals Calculator - Simplify Radicals
Multiply two radical expressions of the form a√x and b√y and get the fully simplified result. Automatically factors out perfect squares.
Multiplying Radicals Calculator
Enter the coefficients and radicands of your two radical expressions to compute and simplify their product.
First Radical (a√x)
Second Radical (b√y)
About the Multiplying Radicals Calculator
A radical expression contains a radical sign (√) applied to a radicand. The square root √x represents the non-negative number whose square equals x. Multiplying two radical expressions combines the radical product rule with coefficient arithmetic to produce a simplified result.
The radical product rule states that √a × √b = √(a×b) for any non-negative real numbers a and b. When the radicals have coefficients, the full product rule for expressions a√x and b√y is: a√x × b√y = (a×b)√(x×y). The outer coefficients multiply together, and the radicands multiply together under a single radical sign.
After multiplying, the result is simplified by factoring out any perfect square from the combined radicand. A perfect square is an integer that is the square of another integer: 1, 4, 9, 16, 25, 36, and so on. If the combined radicand equals k² × m where m contains no perfect-square factor greater than 1, then √(k²×m) = k√m, and the k is pulled outside the radical and multiplied into the coefficient. For example, 3√2 × 2√8 = 6√16 = 6×4 = 24, since √16 = 4 and the radicand simplifies to 1.
Special cases arise frequently. When x = y (the two radicands are equal), the product is a√x × b√x = ab√(x²) = ab×x, which is a whole number with no radical sign. This situation underpins the technique of multiplying a radical expression by its conjugate to eliminate radicals from a denominator. When the combined radicand is itself a perfect square, the result is always a whole number.
Multiplying radicals appears throughout mathematics and physics. In geometry, the diagonal of a rectangle with sides √a and √b is found by the Pythagorean theorem, which involves multiplying and simplifying radicals. In quadratic equations, the discriminant √(b²−4ac) often needs to be simplified by factoring. In trigonometry, many exact values of sine and cosine involve products of radicals like √2 and √3. Understanding how to multiply and simplify radicals is essential for clean algebraic manipulation in these and many other contexts.
Multiplying Radicals Examples
Common radical multiplication problems showing the intermediate and simplified forms.
| Expression | Simplified Result | Notes |
|---|---|---|
| 2√3 × 3√3 | 18 | 6√9 = 6×3 = 18; radicands equal |
| 3√2 × 2√8 | 24 | 6√16 = 6×4 = 24; perfect square |
| √5 × √5 | 5 | 1√25 = 5; product is a whole number |
| 2√3 × √12 | 12 | 2√36 = 2×6 = 12 |
How to Use the Calculator
- Enter the coefficient of the first radical expression in 'Coefficient (a)' (use 1 if there is no coefficient).
- Enter the radicand (the number under the square root) of the first expression in 'Radicand (x)'.
- Enter the coefficient and radicand of the second radical expression in the corresponding fields.
- Click Calculate to see the intermediate product and the fully simplified result.
- Click Reset to clear all fields and start a new calculation.
Frequently Asked Questions
What is the radical product rule?
The radical product rule states that √a × √b = √(a×b) for non-negative real numbers a and b. This means you can combine or split radicals by multiplying or factoring under the square root sign. The rule extends to expressions with coefficients: a√x × b√y = (ab)√(xy).
How do you simplify a radical after multiplication?
Factor the combined radicand into a perfect square times a remaining factor. For example, √72 = √(36×2) = 6√2, since 36 is a perfect square and 2 has no perfect square factor. The calculator finds the largest perfect square divisor automatically.
What if the two radicands are the same?
When x = y, the product a√x × b√x = ab√(x²) = ab×x, which is always a whole number (assuming integer inputs). For instance, 5√7 × 3√7 = 15√49 = 15×7 = 105. This identity is used when rationalising denominators.
Can I enter decimal radicands?
The calculator accepts any non-negative number as a radicand and computes the result numerically. For the simplification step to work cleanly, integer radicands are recommended because the perfect-square factoring algorithm operates on integers.
What does it mean when the result has no radical sign?
When the combined radicand is a perfect square, its square root is an integer, so the entire result simplifies to a whole number with no radical. This happens when both radicands are equal, when their product is a perfect square (such as 4 × 9 = 36), or when the combined radicand is 1.