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Simplify Radicals Calculator

Simplify square roots, cube roots, and nth roots with step-by-step solutions. Learn the prime factorization method and perform radical operations. 100% free.

šŸ’”

āˆš72 = 6āˆš2 ā‰ˆ 8.485

Because 72 = 36 Ɨ 2 = 6Ā² Ɨ 2, so āˆš72 = āˆš(6Ā² Ɨ 2) = 6āˆš2

āˆš Enter Radical

āˆš

šŸ“Š Simplified Form

āˆš72 =

6āˆš2

ā‰ˆ 8.4853

šŸ“ Step-by-Step Solution

1Step 1: Prime factorization of 72 = 2Ā³ Ɨ 3Ā²
2Step 2: Group prime factors into pairs (for square root)
3Step 3: Extract 2 Ɨ 3 = 6 outside the radical
4Step 4: 2 = 2 remains inside
5Result: āˆš72 = 6āˆš2

šŸ“Š Perfect Squares & Cubes Reference

Perfect Squares (nĀ²)

1Ā²
1
2Ā²
4
3Ā²
9
4Ā²
16
5Ā²
25
6Ā²
36
7Ā²
49
8Ā²
64
9Ā²
81
10Ā²
100
11Ā²
121
12Ā²
144
13Ā²
169
14Ā²
196
15Ā²
225

Perfect Cubes (nĀ³)

1Ā³
1
2Ā³
8
3Ā³
27
4Ā³
64
5Ā³
125
6Ā³
216
7Ā³
343
8Ā³
512
9Ā³
729
10Ā³
1000

āˆš How to Simplify Radicals

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

The Prime Factorization Method

The most reliable way to simplify radicals is using prime factorization:

  1. Factor the radicand into its prime factors
  2. Group the factors based on the root index (pairs for āˆš, triplets for āˆ›)
  3. Extract one factor from each complete group to outside the radical
  4. Multiply the outside factors together and inside factors together

Example: Simplify āˆš72

  1. 72 = 2 Ɨ 2 Ɨ 2 Ɨ 3 Ɨ 3 = 2Ā³ Ɨ 3Ā²
  2. Group into pairs: (2 Ɨ 2) Ɨ 2 Ɨ (3 Ɨ 3)
  3. Extract one 2 and one 3: 2 Ɨ 3 = 6
  4. Remaining inside: 2
  5. Result: āˆš72 = 6āˆš2

Rules for Radical Operations

  • Multiplication: āˆša Ɨ āˆšb = āˆš(ab)
  • Division: āˆša Ć· āˆšb = āˆš(a/b)
  • Addition/Subtraction: Only combine "like radicals" with the same radicand
  • Power Rule: (āˆša)Ā² = a

Common Mistakes to Avoid

  • āŒ āˆš(a + b) ā‰  āˆša + āˆšb (cannot split addition under radical)
  • āŒ Forgetting to check for more perfect square factors
  • āŒ Adding radicals with different radicands
  • āœ… Always simplify radicals before adding or subtracting

šŸ“ Common Simplifications

āˆš8 = 2āˆš2

āˆš12 = 2āˆš3

āˆš18 = 3āˆš2

āˆš50 = 5āˆš2

āˆš72 = 6āˆš2

āˆš98 = 7āˆš2

āˆš128 = 8āˆš2

šŸ“ Key Formulas

āˆš(ab) = āˆša Ɨ āˆšb

āˆš(a/b) = āˆša / āˆšb

(āˆša)Ā² = a

āˆšaĀ² = |a|

āæāˆš(aāæ) = a

Frequently Asked Questions

To simplify a radical: 1) Find the prime factorization of the number under the radical. 2) Group the prime factors based on the root index (pairs for square root, triplets for cube root). 3) For each complete group, move one factor outside the radical. 4) Multiply the factors outside together and the remaining factors inside together. For example, āˆš72 = āˆš(2Ā³ Ɨ 3Ā²) = āˆš(2Ā² Ɨ 2 Ɨ 3Ā²) = 2 Ɨ 3 Ɨ āˆš2 = 6āˆš2.

āˆš72 = 6āˆš2. Here's how: 72 = 36 Ɨ 2 = 6Ā² Ɨ 2. Since 36 is a perfect square, āˆš72 = āˆš(36 Ɨ 2) = āˆš36 Ɨ āˆš2 = 6āˆš2. You can verify: (6āˆš2)Ā² = 36 Ɨ 2 = 72. The decimal approximation is approximately 8.485.

For variables under a radical, use the same grouping principle. For square roots, pair the exponents. For example: āˆš(xā“) = xĀ² (because xā“ = (xĀ²)Ā²), āˆš(xāµ) = xĀ²āˆšx (because xāµ = xā“ Ɨ x = (xĀ²)Ā² Ɨ x), āˆš(xā¶yā“) = xĀ³yĀ² (because xā¶ = (xĀ³)Ā² and yā“ = (yĀ²)Ā²). For cube roots, group exponents in threes.

You can only add or subtract radicals that have the same radicand (the number under the radical) AND the same index (square root, cube root, etc.). For example: 3āˆš2 + 5āˆš2 = 8āˆš2 (same radicand). But āˆš2 + āˆš3 cannot be simplified further because the radicands are different. Sometimes you need to simplify first: āˆš8 + āˆš2 = 2āˆš2 + āˆš2 = 3āˆš2.

To multiply radicals with the same index, multiply the radicands together under one radical, then simplify. Formula: āˆša Ɨ āˆšb = āˆš(a Ɨ b). For example: āˆš2 Ɨ āˆš8 = āˆš(2 Ɨ 8) = āˆš16 = 4. With coefficients: 3āˆš2 Ɨ 4āˆš5 = (3 Ɨ 4) Ɨ āˆš(2 Ɨ 5) = 12āˆš10. For different indices, convert to fractional exponents first.

āˆš (square root) finds a number that when multiplied by itself gives the radicand: āˆš9 = 3 because 3 Ɨ 3 = 9. āˆ› (cube root) finds a number that when multiplied by itself THREE times gives the radicand: āˆ›8 = 2 because 2 Ɨ 2 Ɨ 2 = 8. Square roots need pairs of factors to simplify; cube roots need triplets. For example: āˆš8 = 2āˆš2 (one pair of 2s), but āˆ›8 = 2 (one triplet of 2s).

To rationalize a denominator with a single radical, multiply both numerator and denominator by that radical. For example: 1/āˆš2 = (1 Ɨ āˆš2)/(āˆš2 Ɨ āˆš2) = āˆš2/2. For denominators like (a + āˆšb), multiply by the conjugate (a - āˆšb). For example: 1/(1 + āˆš2) = (1 - āˆš2)/((1 + āˆš2)(1 - āˆš2)) = (1 - āˆš2)/(1 - 2) = (1 - āˆš2)/(-1) = āˆš2 - 1.

āˆš Disclaimer: This calculator is for educational purposes. Always verify your work, especially for homework and exams. The calculator handles positive integers; for variables and complex expressions, consult your textbook or instructor.