Rationalize the Denominator

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

Expression

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

Numerator

Denominator

Our rationalize the denominator calculator helps you rationalize the denominators containing radicals. Moreover, the tool can also rationalize expressions with complex square root terms as well.

How To Rationalize The Denominator And Simplify?

When it comes to rationalizing denominators with radicals, below are the four possibilities, and these are also used by our rationalize the denominator calculator.

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

Multiply the given expression with the term below:

ᵏ√(y^ᵏ⁻¹) / ᵏ√(y^ᵏ⁻¹)

By finding the product, you will get the answer in the form:

x * ᵏ√y * ᵏ√(y^ᵏ⁻¹)

= x * ᵏ√(y^ᵏ)

= x * y

This is the most simple case that this rationalize denominator calculator works on to generate accurate results.

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

Find the product of the given term with the ᵏ√(y^ᵏ⁻¹) / ᵏ√(y^ᵏ⁻¹) as same for the first case
After that, the most important factor that this calculator considers for accurate calculations is a product of the quantity ᵏ√(y^ᵏ⁻¹) with both numerator monomials separately

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

This is where the actual technicality begins!

Go by multiplying the denominator terms with (x * √y - z * √u) / (x * √y - z * √u)
After that, you should strive to get the simplified term in the form of the following formula:

(a - b) * (a + b) = a^2 - b^2

This will lead you to get the simplification of the rationalizing denominators in the final form in the denominator as follows:

x^2 - y^2

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

As we are coping with the manual formulas here, so you need to multiply both the quantities in the numerator by the following expression separately:

(x * √y - z * √u ) /(x * √y - z * √u)

How to Rationalize the Denominator?

Let’s resolve an example to clarify your concept regarding rationalizing denominators!

Example # 01:

How do you rationalize a denominator given as under:

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

Solution:

Here we have

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

$$ \frac{3 * \sqrt{5}}{4 * \sqrt{4*4}} $$

$$ \frac{3 * \sqrt{5}}{4 * \sqrt{4^{2}}} $$

$$ \frac{3 * \sqrt{5}}{4 * 4} $$

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

$$ 0.1875 * \sqrt{5} $$

This is the required answer that can also be verified by this rationalize denominator calculator.

Working of Rationalize The Denominator Calculator:

The rationalize calculator is loaded with a simple user-interface that lets you enter a few inputs to get instant rationalization of the denominators.

Input:

First of all, select the mode i.e; “Simple” or “Advanced”

If You Select Simple Mode:

Select the expression from the drop-down list
Now enter the required parameters of the expressions in their designated fields
Tap Calculate

If You Select Advanced Mode:

Enter the numerator and denominator expressions
Tap Calculate

Output:

Rationalization of denominators
Step-by-step calculations

Faqs:

Should You Always Rationalize The Denominator?

No, of course not! Rationalizing is only done when you are stuck with complicated calculations and there seems no solution to simplify the problem.

What Should Be Multiplied And Divided To Rationalise The Denominator of 1 √ 7?

To rationalize the given denominator, go by multiplying the expression with √7/√7

such that:

1/√7 x √7/√7

= √7

References:

From the source Wikipedia: Rationalisation (mathematics), Rationalisation of a monomial square root and cube root, Dealing with more square roots, Generalizations From the source Lumen Learning: Rationalize Denominators, One Term, Two Terms