Radical Expression Calculator
Simplify, add, subtract, multiply, and divide radical expressions with step-by-step solutions
About Radical Expressions
A radical expression contains a root symbol (√ or ∛). The number under the radical is called the radicand, and the small number indicating the root is the index (2 for square root, 3 for cube root, etc.).
Operations on Radicals
- Simplify: Factor out perfect squares/cubes from the radicand
- Add/Subtract: Only like radicals (same index and radicand) can be combined
- Multiply: Multiply coefficients together and radicands together: √a × √b = √(ab)
- Divide: Divide coefficients and radicands separately, then rationalize if needed
Frequently Asked Questions
What is a radical expression?
A radical expression is a mathematical expression that contains a root symbol (radical). It includes square roots (√), cube roots (∛), and higher-order roots. For example, 2√12 or 5∛16.
How do you simplify a radical?
To simplify a radical, find the largest perfect square (or perfect cube for cube roots) that divides the radicand. Factor it out and take its root. For example, √72 = √(36 × 2) = 6√2.
Can you add radicals with different radicands?
No, you can only add or subtract like radicals - those with the same index and radicand. First simplify each radical, then check if they have the same radicand before combining.
How do you multiply radicals?
Multiply the coefficients together and the radicands together. For example, (2√3)(4√5) = 8√15. The radicals must have the same index to multiply directly.
What does it mean to rationalize a denominator?
Rationalizing means removing radicals from the denominator by multiplying both numerator and denominator by an appropriate radical expression. This creates an equivalent fraction with no radicals in the denominator.
What is the difference between √ and ∛?
The √ symbol represents a square root (index 2), while ∛ represents a cube root (index 3). Square roots find what number squared gives the radicand, while cube roots find what number cubed gives the radicand.
Can radicals be negative?
For square roots (even index), the radicand must be non-negative in real numbers. For cube roots (odd index), the radicand can be negative. The coefficient in front of a radical can always be negative.
How do you check if a radical is fully simplified?
A radical is fully simplified when: (1) no perfect square factors remain under a square root (or perfect cube factors under a cube root), (2) no fractions are under the radical, and (3) no radicals appear in the denominator.
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