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Rationalize the Denominator Calculator

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:


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

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

$$ \frac{ a\sqrt{b} }{ x\sqrt{y}+z\sqrt{u} } \ = \ ?$$

$$ \frac{ a\sqrt{b}+c\sqrt{d} }{ x\sqrt{y}+k\sqrt{u} } \ = \ ?$$













Enter Numerator:

Enter Denominator:


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Use this online rationalize the denominator calculator to know how to rationalize the denominators containing radicals.

Let’s get to the point of discussion which is rationalizing denomiantors with complex square root terms and is explained with suitable examples in the article below.

Stay Focused!

What Is Rationalizing Denominators?

In the context of mathematics:

“A particular technique to vanish square or cube root in the denominator of a fraction is known as the simplify by rationalizing the denominators”.

How to Rationalize the Denominator and Simplify the Results?

When it comes to rationalizing denominators with radicals, we are left with four possibilities that are also dealt by this online rationalize the denominator and simplify calculator. Among these are:

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

This is the most simple case that this rationalize denominator calculator works on to generate accurate results. Go by following the guide below to reduce these type of expressions:

  • 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

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

In this case, we have an addition which is the sum of two monomials or say quantities in the numerator. Simplification in such a case requires the following instructions to be tackled:

  • 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 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! Now you are about to deal with a couple of quantities that need to be rationalised. If you utilise this calculator, the calculation becomes ultimately accurate and swift enough. But throwing a light on the manual process is also important. You should keep in mind the steps mentioned as under to complete the computations:

  • 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)):

The case is exactly the same as of the one aforementioned and can be better resolved by utilising this rationalize the denominator calculator.

  • 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 Do You Rationalize The Denominator?

Let’s resolve a couple of examples to clarify your concept regarding rationalizing denominators!

Example # 01:

How do you rationalize a denominator given as under:

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


Here we have:

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

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

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

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

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

$$ 0.1875 * \sqrt{5} $$

Which is the required answer.

Example # 02:

How to rationalize a denominator expression given as follows:

$$ \frac{3 * \sqrt{9}}{4 * [3]\sqrt{25}} $$


$$ \frac{3 * \sqrt{9}}{4 * \sqrt[3]{25}} $$

$$ \frac{3 * \sqrt{9}}{4 * \sqrt[3]{25}} * \frac{\sqrt[3]{\left(25\right)^{2}}}{\sqrt[3]{\left(25\right)^{2}}} $$

$$ \frac{3 * 3 * \sqrt[6]{390625}}{4*25} $$

$$ = 0.09 * \sqrt[6]{625} $$

$$ = 0.09 * \sqrt[3]{25} $$

You can also verify the results by the online rationalize denominator calculator.

How Does Rationalize The Denominator Calculator Work?

Go by following the guide below to utilise the free rational denominator calculator.


  • First of all, select the mode that can be either “Simple” or “Advance”

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
  • At last, hit the calculate button

If You Select Advanced Mode:

  • Enter the numerator and denominator expressions in the respective fields and hit the calculate button


The free rationalizing the denominator calculator with square roots does the following calculations:

  • Simplify the rationalizing denominators
  • Display step by step calculations


What Is Rationalisation Class 9?

Rationalization is a process in which you are required to remove the square roots of the denominator terms by multiplying the whole expression with an irrational number. Most of the time, the square root terms do come in radicals rationalization that could better be resolved by using the rationalize the denominator calculator with square roots in a couple of moments.

What Will be Get After Rationalize Denominator and Simplify of 1/√5 √2?

The simplest radical form with a rational denominator calculator for the given problem is 1/√10.

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

Why Do We Rationalize the Numerator?

When there is no need of any radical term in the numerator of the fraction and in actual it is there. Then you need to rationalize the fraction in order to remove the radical term involved in the expression. For more insurance, you may subject to this online rationalize the denominator calculator for better estimation of the results.

Should You Always Rationalise 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. Usually, the radicals are noted most of the time in trigonometric expressions, and this is where our free rationalize the denominator and simplify calculator comes into play while giving you simplified rationalised results.


The concept of rationalization introduces a simplicity while coping with complex fraction terms containing radicals. And due to the accurate and fast calculations provided by our free rationalize the denominator calculator with square roots in this regard, pupils and professionals do rely on our tool to attain accuracy in the outcomes.


From the source of Wikipedia: Rationalisation (mathematics), Rationalisation of a monomial square root and cube root, Dealing with more square roots, Generalizations

From the source if Lumen Learning: Rationalize Denominators, One Term, Two Terms