Add & Subtract Radicals Calculator
Combine like radicals with step-by-step simplification and detailed solutions
Common Examples
2√12 + 5√3 = 9√3
6√8 − 3√2 = 9√2
√50 + √18 = 8√2
4√27 − √75 = 7√3
Adding and Subtracting Radicals
Adding and subtracting radicals is similar to combining like terms in algebra. You can only combine radicals that have the same index and the same radicand (the number under the radical).
Rules for Combining Radicals
- Simplify first: Always simplify each radical before attempting to combine
- Check for like radicals: Radicals must have the same index and radicand
- Add/subtract coefficients: If they're like radicals, add or subtract the coefficients
- Keep the radical: The radical part stays the same
Example
Add: 2√12 + 5√3
- Simplify 2√12 = 2√(4×3) = 2(2)√3 = 4√3
- 5√3 is already simplified
- Both have √3, so combine: 4√3 + 5√3 = 9√3
Frequently Asked Questions
What are like radicals?
Like radicals are radical expressions that have the same index and the same radicand. For example, 3√5 and 7√5 are like radicals because they both have index 2 and radicand 5. Only like radicals can be combined by adding or subtracting.
Can you add √2 and √3?
No, √2 and √3 cannot be added together to form a single radical because they have different radicands. The expression √2 + √3 is already in its simplest form.
Do I need to simplify before adding radicals?
Yes! Always simplify each radical first. Sometimes radicals that look different are actually like radicals after simplification. For example, √8 + √18 becomes 2√2 + 3√2 = 5√2 after simplifying.
How do you subtract radicals?
Subtracting radicals follows the same rules as adding. First simplify each radical, verify they're like radicals, then subtract the coefficients. For example: 5√3 − 2√3 = (5 − 2)√3 = 3√3.
Can you add a square root and a cube root?
No, you cannot combine radicals with different indices. A square root (√) and a cube root (∛) have different indices and cannot be added or subtracted, even if they have the same radicand.
What if the coefficients become zero?
If subtracting results in a coefficient of zero, the answer is simply 0. For example, 3√5 − 3√5 = (3 − 3)√5 = 0√5 = 0.
How do you combine multiple radicals?
When combining more than two radicals, simplify all terms first, group like radicals together, then combine the coefficients of each group. For example: √8 + √18 − √2 = 2√2 + 3√2 − √2 = 4√2.
Can the coefficient be a fraction?
Yes, coefficients can be fractions. When adding or subtracting, treat the fractional coefficients like any other fractions. For example: (1/2)√3 + (1/3)√3 = (5/6)√3.
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