Radical Calculator
Add, subtract, multiply, and divide radicals
Understanding Radical Operations
Working with radicals requires understanding when you can combine them and when you must keep them separate. The key is recognizing like and unlike radicals.
Adding and Subtracting Radicals
Rule: You can only add/subtract radicals with the same radicand (like radicals)
Think of it like combining like terms: just as 2x + 3x = 5x, we have 2√3 + 3√3 = 5√3
Multiplying Radicals
When multiplying radicals:
- Multiply the coefficients
- Multiply the radicands
- Simplify the result if possible
- Example: 2√3 × 5√6 = 10√18 = 10(3√2) = 30√2
Dividing Radicals
When dividing radicals:
- Divide the coefficients
- Divide the radicands (or write as a single radical)
- Rationalize the denominator if needed
- Simplify the result
Key Properties
- √a × √b = √(a × b)
- √a ÷ √b = √(a ÷ b)
- a√c + b√c = (a + b)√c
- (√a)² = a
Frequently Asked Questions
Can I add radicals with different radicands?
No, you cannot combine radicals with different radicands through addition or subtraction. √2 + √3 cannot be simplified further - they must remain as separate terms. Only like radicals (same radicand) can be added or subtracted by combining their coefficients.
How do I multiply radicals with different radicands?
Unlike addition, you can multiply radicals with different radicands. Simply multiply the coefficients together and the radicands together. For example: 2√3 × 5√2 = (2×5)√(3×2) = 10√6. Always check if the result can be simplified further.
What does "like radicals" mean?
Like radicals are radical expressions that have the same index (like square root) and the same radicand (number under the radical). For example, 2√5 and 7√5 are like radicals, but 2√5 and 2√3 are not, and neither are √5 and ∛5.
Should I simplify radicals before operating on them?
Yes, it's usually best to simplify each radical first. This can reveal like radicals that weren't obvious initially. For example, √8 + √18 looks like unlike radicals, but when simplified to 2√2 + 3√2, you can see they're like radicals that combine to give 5√2.
How do I handle negative coefficients with radicals?
Treat negative coefficients just like negative numbers in regular algebra. For subtraction, change the sign of the second term's coefficient: 5√3 - 2√3 = (5-2)√3 = 3√3. For multiplication, apply the standard rules for multiplying signed numbers.
Can radicals equal zero?
Yes, when the coefficient is zero or when subtracting equal radicals. For example, 3√2 - 3√2 = 0, or 0√5 = 0. The radicand √0 also equals 0. However, you cannot have a negative number under a square root (in real numbers).