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Radical Calculator

Simplify radical expressions, evaluate roots, and perform operations with radicals. Get step-by-step solutions to understand the simplification process.

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Quick Example: √72 = 6√2

72 = 36 × 2 = 6² × 2, so √72 = √(6² × 2) = 6√2

✨ Simplify Radical Expression

√

Try these examples:

📊 Simplified Result

√72 =

6√2

≈ 8.4853

Step-by-Step Solution:

1Prime factorization of 72: 2 × 2 × 2 × 3 × 3
2Group factors in pairs
3Extract: 2 × 3 = 6
4Remaining under radical: 2 = 2
5Result: 6√2

√ How to Simplify Radical Expressions

Simplifying radicals is a fundamental skill in algebra. The goal is to rewrite a radical expression in its simplest form by extracting perfect square (or cube, etc.) factors from under the radical sign.

Step-by-Step Method

  1. Find prime factorization of the number under the radical
  2. Group factors based on the index (pairs for √, triplets for ∛)
  3. Extract complete groups outside the radical
  4. Multiply remaining factors inside the radical

Example: Simplify √72

√72

= √(2 × 2 × 2 × 3 × 3)     ← prime factorization

= √(2² × 3² × 2)     ← group in pairs

= 2 × 3 × √2     ← extract pairs

= 6√2 ✓

Perfect Squares to Know

n12345678910
n²149162536496481100

Radical Operation Rules

Multiplication

√a × √b = √(ab)

Division

√a ÷ √b = √(a/b)

Addition (same radicand)

a√c + b√c = (a+b)√c

Power Rule

√a = a^(1/2)

📋 Common Simplifications

√8 = 2√2

√12 = 2√3

√18 = 3√2

√32 = 4√2

√48 = 4√3

√50 = 5√2

√72 = 6√2

√98 = 7√2

🧊 Perfect Cubes

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Frequently Asked Questions

To simplify a radical: 1) Find the prime factorization of the number under the radical. 2) Group the factors based on the index (pairs for square roots, triplets for cube roots). 3) Move complete groups outside the radical. 4) Multiply remaining factors inside. For example, √72 = √(36 × 2) = √36 × √2 = 6√2.

√32 simplifies to 4√2. Here's why: 32 = 16 × 2, and √16 = 4. So √32 = √(16 × 2) = √16 × √2 = 4√2. The decimal value is approximately 5.657.

A radical can be simplified if the number under it has a perfect square factor (for square roots) or perfect cube factor (for cube roots). Check if the radicand is divisible by 4, 9, 16, 25, 36, etc. If yes, you can simplify it.

√ (square root) asks 'what number times itself equals this?' For example, √9 = 3 because 3 × 3 = 9. ³√ (cube root) asks 'what number times itself three times equals this?' For example, ³√8 = 2 because 2 × 2 × 2 = 8.

You can only add or subtract radicals with the same radicand (number under the radical). For example: 3√5 + 2√5 = 5√5. But 3√5 + 2√3 cannot be combined because the radicands (5 and 3) are different. This is similar to combining like terms in algebra.

To multiply radicals with the same index, multiply the numbers under the radicals: √a × √b = √(ab). For example: √3 × √12 = √36 = 6. If there are coefficients, multiply them separately: 2√3 × 4√5 = 8√15.

⚠️ Disclaimer: This calculator simplifies radical expressions and performs basic radical operations. For complex equations involving radicals (like √(x+3) = 5), please use a computer algebra system or consult your textbook for solving techniques.