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Simplify Algebra Cube Root Calculator

Reviewed by Calculator Editorial Team

Simplifying cube roots is a fundamental algebra skill that helps you work with radicals more efficiently. This calculator will help you simplify cube roots of integers and variables step-by-step.

What is a Cube Root?

The cube root of a number x is a value that, when multiplied by itself three times, gives the original number. In mathematical terms:

If \( y = \sqrt[3]{x} \), then \( y^3 = x \)

For example, the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \). Cube roots can be simplified when the radicand (the number inside the root) contains perfect cubes.

How to Simplify Cube Roots

To simplify a cube root, follow these steps:

  1. Factor the radicand into perfect cubes and other factors.
  2. Separate the perfect cube factors from the other factors.
  3. Take the cube root of the perfect cube factors.
  4. Leave the remaining factors inside the cube root.

General form: \( \sqrt[3]{a \times b^3} = b \times \sqrt[3]{a} \)

For example, to simplify \( \sqrt[3]{72} \):

  1. Factor 72: \( 72 = 8 \times 9 \)
  2. 8 is a perfect cube (\( 2^3 \))
  3. Take the cube root of 8: \( \sqrt[3]{8} = 2 \)
  4. Final simplified form: \( 2\sqrt[3]{9} \)

Examples

Example 1: Simplifying \( \sqrt[3]{54} \)

  1. Factor 54: \( 54 = 27 \times 2 \)
  2. 27 is a perfect cube (\( 3^3 \))
  3. Take the cube root of 27: \( \sqrt[3]{27} = 3 \)
  4. Final simplified form: \( 3\sqrt[3]{2} \)

Example 2: Simplifying \( \sqrt[3]{162} \)

  1. Factor 162: \( 162 = 81 \times 2 \)
  2. 81 is not a perfect cube, but 162 can be factored further: \( 162 = 27 \times 6 \)
  3. 27 is a perfect cube (\( 3^3 \))
  4. Take the cube root of 27: \( \sqrt[3]{27} = 3 \)
  5. Final simplified form: \( 3\sqrt[3]{6} \)

Example 3: Simplifying \( \sqrt[3]{x^6y^3} \)

  1. Identify perfect cubes: \( x^6 \) and \( y^3 \)
  2. Take the cube root of each: \( \sqrt[3]{x^6} = x^2 \) and \( \sqrt[3]{y^3} = y \)
  3. Final simplified form: \( x^2y \)

Common Mistakes

Be careful not to:

  • Take the cube root of each factor separately (e.g., \( \sqrt[3]{12} = \sqrt[3]{3} \times \sqrt[3]{4} \) is incorrect)
  • Assume all numbers can be simplified (e.g., \( \sqrt[3]{17} \) cannot be simplified)
  • Forget to simplify the entire radicand (e.g., \( \sqrt[3]{24} = \sqrt[3]{8} \times \sqrt[3]{3} \) is correct, but \( \sqrt[3]{8} \times \sqrt[3]{3} \) is not simplified)

FAQ

Can cube roots be simplified if the radicand is negative?
Yes, but the result will be negative. For example, \( \sqrt[3]{-8} = -2 \).
What if the radicand has a variable with an exponent that's not a multiple of 3?
You can only simplify the part that forms a perfect cube. For example, \( \sqrt[3]{x^5} = x\sqrt[3]{x^2} \).
How do I simplify cube roots with decimals?
Convert the decimal to a fraction first, then simplify as usual. For example, \( \sqrt[3]{0.125} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \).