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Rationalize The Denominator and Simplify Cube Root Calculator

Reviewed by Calculator Editorial Team

This calculator helps you rationalize denominators containing cube roots and simplify the resulting expressions. Learn the step-by-step process with clear examples and formulas.

What is Rationalizing the Denominator?

Rationalizing the denominator means eliminating any radicals (like square roots or cube roots) from the denominator of a fraction. This process makes expressions easier to work with and compare.

For cube roots, we typically multiply both the numerator and denominator by a form of 1 that will eliminate the cube root in the denominator. The most common approach is to multiply by the cube root of the denominator's radicand raised to the power of 2.

For a denominator of the form ∛a, multiply numerator and denominator by ∛(a²):

∛a = a^(1/3)

∛(a²) = a^(2/3)

∛a × ∛(a²) = a^(1/3 + 2/3) = a^1 = a

This process works because when you multiply a cube root by itself twice, you're essentially raising it to the power of 3, which cancels out the cube root in the denominator.

Simplifying Cube Roots

Simplifying cube roots involves expressing the radicand (the number inside the cube root) as a product of perfect cubes and other factors. A perfect cube is any number that can be expressed as n³ where n is an integer.

∛(a × b) = ∛a × ∛b

∛(a³ × b) = a × ∛b

For example, ∛(27x) can be simplified to 3∛x because 27 is a perfect cube (3³).

Steps to Simplify Cube Roots

  1. Factor the radicand into perfect cubes and other factors.
  2. Separate the cube roots of the perfect cubes from the other factors.
  3. Multiply the cube roots of the perfect cubes together.

Combined Process

When you have a fraction with a cube root in the denominator, you'll typically need to both rationalize the denominator and simplify the cube roots. Here's the complete process:

Step-by-Step Process

  1. Identify the radicand in the denominator.
  2. Multiply numerator and denominator by the cube root of the radicand raised to the power of 2.
  3. Simplify the resulting cube roots in both numerator and denominator.
  4. Combine any like terms and simplify the final expression.

Remember: You can only rationalize denominators with cube roots if the radicand is a perfect cube or can be expressed in terms of perfect cubes.

Worked Examples

Example 1: Simple Rationalization

Rationalize the denominator of 1/∛2.

Solution:

  1. Multiply numerator and denominator by ∛(2²):
  2. 1/∛2 × ∛(2²)/∛(2²) = ∛(2²)/(2)
  3. Simplify ∛(2²):
  4. ∛(2²) = 2^(2/3)
  5. Final expression: 2^(2/3)/2

Example 2: Combined Process

Simplify and rationalize 1/(∛3 + ∛2).

Solution:

  1. Multiply numerator and denominator by (∛3 - ∛2):
  2. 1/(∛3 + ∛2) × (∛3 - ∛2)/(∛3 - ∛2) = (∛3 - ∛2)/(3 - 2)
  3. Simplify the denominator:
  4. (∛3 - ∛2)/1 = ∛3 - ∛2

FAQ

Can I rationalize denominators with cube roots that aren't perfect cubes?

Yes, but the result will still contain a cube root in the denominator. The process is the same, but the final expression won't be completely rationalized.

What if the denominator has multiple cube roots?

You would multiply by the product of each cube root raised to the power of 2. For example, for ∛a + ∛b, you would multiply by (∛a + ∛b)(∛a² - ∛a∛b + ∛b²).

Is there a difference between rationalizing denominators with square roots and cube roots?

Yes. For square roots, you multiply by the same square root. For cube roots, you multiply by the cube root raised to the power of 2.