Combine and Simplify The Following Radical Expression Calculator
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Reviewed by Calculator Editorial Team
This calculator helps you combine and simplify radical expressions by following mathematical rules for radicals. Whether you're studying algebra or need to solve a problem, this tool provides step-by-step guidance and verification of your results.
How to Use This Calculator
To combine and simplify radical expressions using this calculator:
- Enter the radical expressions you want to combine in the input fields.
- Select the operation (addition, subtraction, multiplication, or division).
- Click "Calculate" to see the simplified result.
- Review the step-by-step solution provided.
The calculator follows these fundamental rules for radicals:
- √(a) × √(b) = √(a × b)
- √(a) + √(a) = 2√(a)
- √(a/b) = √(a)/√(b)
- √(a²) = a (when a ≥ 0)
Formula Used
Combining Radicals Formula
When combining radicals with the same index and radicand, you can use the following formulas:
- √a + √a = 2√a
- √a - √a = 0
- √a × √b = √(a × b)
- √a / √b = √(a/b)
For more complex expressions, the calculator applies these rules systematically to simplify the expression to its most reduced form.
Worked Examples
Example 1: Adding Like Radicals
Problem: Simplify √8 + √2
Solution:
- √8 can be simplified to 2√2 (since 8 = 4 × 2 and √4 = 2)
- Now we have 2√2 + √2
- Combine like terms: (2 + 1)√2 = 3√2
Final simplified form: 3√2
Example 2: Multiplying Radicals
Problem: Simplify √6 × √3
Solution:
- Multiply the radicands: 6 × 3 = 18
- Take the square root of the product: √18
- Simplify √18 to 3√2 (since 18 = 9 × 2 and √9 = 3)
Final simplified form: 3√2
Frequently Asked Questions
What is the difference between like radicals and unlike radicals?
Like radicals have the same radicand (the number under the radical) and the same index (the number in front of the radical). Unlike radicals have different radicands or indices. Like radicals can be combined, while unlike radicals cannot be combined directly.
How do I simplify a radical expression with a coefficient?
To simplify a radical with a coefficient, factor the radicand into a perfect square and another factor. For example, √18 = √(9 × 2) = √9 × √2 = 3√2.
Can I combine radicals with different indices?
No, you can only combine radicals with the same index. For example, √2 + ³√3 cannot be combined because they have different indices (2 and 3).