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Rationalize the Denominator Calculator

Enter the numerator and denominator terms of the radicals, and the tool will rationalize them to the simplest radical form, with the steps shown.

To Calculate:

Simple

Advance

radical

$ \frac{a\sqrt[n]b}{x\sqrt[k]y} \ = \ ? $

Denominator

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Rationalize Denominator Calculator

Our rationalize denominator calculator helps simplify denominators containing radicals. It handles expressions with complex square root terms and provides accurate, step-by-step rationalized results.

How to Rationalize the Denominator

There are four main types of rationalization that our calculator can handle efficiently:

**1. Radical / Radical ((a * ⁿ√b) / (x * ᵏ√y))**

Multiply the fraction by: ᵏ√(yᵏ⁻¹) / ᵏ√(yᵏ⁻¹)
Denominator becomes: x * ᵏ√y * ᵏ√(yᵏ⁻¹) = x * ᵏ√(yᵏ) = x * y

**2. Sum / Radical ((a * ⁿ√b + c * ᵐ√d) / (x * ᵏ√y))**

Multiply numerator and denominator by ᵏ√(yᵏ⁻¹) / ᵏ√(yᵏ⁻¹)
Distribute the radical to each term in the numerator

**3. Radical / Sum ((a * √b) / (x * √y + z * √u))**

Multiply numerator and denominator by the conjugate: (x * √y - z * √u) / (x * √y - z * √u)
Simplify denominator using: (a + b)(a - b) = a² - b²
Denominator becomes: x²y - z²u

**4. Sum / Sum ((a * √b + c * √d) / (x * √y + z * √u))**

Multiply numerator and denominator by the conjugate of the denominator separately:

(x * √y - z * √u) / (x * √y - z * √u)

Example: Rationalizing a Denominator

Example #01:

Rationalize the denominator:

$$ \frac{3 \cdot \sqrt{5}}{4 \cdot \sqrt{16}} $$

Solution:

Step 1: Factor the radical in the denominator:

$$ \frac{3 \cdot \sqrt{5}}{4 \cdot \sqrt{16}} = \frac{3 \cdot \sqrt{5}}{4 \cdot \sqrt{4 \cdot 4}} = \frac{3 \cdot \sqrt{5}}{4 \cdot \sqrt{4^2}} $$

Step 2: Simplify the square root:

$$ \frac{3 \cdot \sqrt{5}}{16} $$

Step 3: Optional decimal representation:

$$ 0.1875 \cdot \sqrt{5} $$

This is the simplified, rationalized result. You can verify it using our calculator.

How the Rationalize Denominator Calculator Works

Input:

Select mode: Simple or Advanced

Simple Mode:

Choose the expression type from the drop-down list
Enter the required parameters
Click "Calculate"

Advanced Mode:

Manually enter numerator and denominator expressions
Click "Calculate"

Output:

Rationalized denominator
Step-by-step solution

FAQs

Should you always rationalize the denominator?

No. Rationalization is only necessary when it simplifies the expression or makes further calculations easier.

How to rationalize 1 / √7?

Multiply numerator and denominator by √7:

1/√7 × √7/√7 = √7 / 7

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