Simplifying Square Roots and Cube Roots Calculator

What Are Square Roots and Cube Roots?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 = 16. Similarly, the cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.

Square roots and cube roots are denoted by the radical symbol (√ for square roots and ∛ for cube roots). For instance, √16 represents the square root of 16, and ∛27 represents the cube root of 27.

How to Simplify Square Roots

Simplifying square roots involves expressing the root in terms of its prime factors. Here are the steps to simplify a square root:

1. Factor the number under the radical into its prime factors.
2. Identify pairs of identical prime factors.
3. Take one number from each pair out of the radical.
4. Multiply the numbers taken out of the radical.
5. If there are any remaining prime factors, leave them under the radical.

For example, to simplify √72:

1. Factor 72 into its prime factors: 72 = 8 × 9 = 2³ × 3²
2. Identify pairs of identical prime factors: 2² and 3²
3. Take one number from each pair out of the radical: √(2² × 3²) = √(2²) × √(3²) = 2 × 3 = 6
4. There are no remaining prime factors, so the simplified form is 6.

How to Simplify Cube Roots

Simplifying cube roots involves expressing the root in terms of its prime factors, similar to square roots but with a focus on groups of three identical factors. Here are the steps to simplify a cube root:

1. Factor the number under the radical into its prime factors.
2. Identify groups of three identical prime factors.
3. Take one number from each group out of the radical.
4. Multiply the numbers taken out of the radical.
5. If there are any remaining prime factors, leave them under the radical.

For example, to simplify ∛108:

1. Factor 108 into its prime factors: 108 = 2 × 54 = 2 × 2 × 27 = 2 × 2 × 3³
2. Identify groups of three identical prime factors: 3³
3. Take one number from each group out of the radical: ∛(2 × 2 × 3³) = ∛(2 × 2) × ∛(3³) = ∛4 × 3 = ∛4 × 3
4. There are no remaining prime factors, so the simplified form is 3∛4.

Examples of Simplifying Roots

Here are some examples of simplifying square roots and cube roots:

Square Root Examples

• √36 = 6
• √80 = √(16 × 5) = 4√5
• √128 = √(64 × 2) = 8√2

Cube Root Examples

• ∛64 = 4
• ∛125 = 5
• ∛192 = ∛(64 × 3) = 4∛3

Common Mistakes to Avoid

When simplifying square roots and cube roots, it's easy to make mistakes. Here are some common errors to avoid:

• Incorrect factorization: Ensure you correctly factor the number under the radical into its prime factors.
• Miscounting pairs or groups: When identifying pairs for square roots or groups of three for cube roots, make sure you count correctly.
• Forgetting to multiply: After taking numbers out of the radical, remember to multiply them together.
• Leaving unnecessary radicals: Only leave prime factors under the radical that cannot be paired or grouped.