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Rationalize the Denominator Calculator

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:


Table of Content


The rationalize the denominator calculator is used to rationalize the denominator for a given input. So rationalizing denominators can be made easy by using our calculator.

What is a Rationalization of Numbers?

“The process of removing imaginary numbers or radicles (square root, cube root) from the denominator of a fraction is known as rationalization”. 

In other words, we say that it is a process of multiplying a surd with another surd to get a rational number. The surd that is used to multiply is called a rationalized factor.  

Standard Form of Rationalization:

As we know fractions have a denominator and numerator therefore we rationalize the fractions to express them in the standard form with the help of rationalize denominator calculator.

Suppose a fractional term is 1/(n-√m), So the rationalized form of the fractional term is written as:

  • [1/( n - √m )] × [( n + √m ) / ( n + √m )]
  • Rationalized Form = [( n + √m ) / ( n2 - m )]

How to Rationalize The Denominator?

The rationalization of the denominator is the process of eliminating the radicles from fractional terms by multiplying the both numerator and denominator by the conjugate of the denominator 

The rule for how do you rationalize the denominator and simplify is the topic for the whole article. In order to get rid of radicles from the denominator and seek rationalizing denominators take the help of our rationalize the denominator calculator and attach to the below points:

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

Multiply the denominator and the numerator by the radicle which will get rid of the radicle from the denominator. Note that when you multiply the denominator and numerator by the exact same thing the fractions will be equivalent. 

  • If the denominator is in the form of √a then multiply the numerator by the denominator √a. 
  • If the denominator is in the form of  n√a^m where m

ᵏ√(b^ᵏ⁻¹) / ᵏ√(b^ᵏ⁻¹)
a * ᵏ√b * ᵏ√(b^ᵏ⁻¹)
= a * ᵏ√(b^ᵏ)
= a * b

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

This rule is also similar to the above discussed but it has two summands in numerator. In this calculation, rationalizing the denominator calculator takes into service the k√(y^k-1) for both numerators. 

Here we again multiply; 

k√(y^k-1) / k√(y^k-1) and obtain x * y.

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

In this case, the summands of both denominators need to be rationalized. For the simplifications of rationalizing the denominators, the [a^2 - b^2 = (a + b)(a - b)] formula is used which gives two squares by eliminating the square root.

So here we multiply our expression;

(x * √y - z * √u) / (x * √y - z * √u)  

(x^2 - y^2) 

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

This point is similar to rule three which has already been discussed multiplying both quantities.  

(x * √y - z * √u) / (x * √y - z * √u). 

Practical Examples:

Rationalize the Denominator with 1 Term:

Suppose a term that is available for how do you rationalize the denominator and simplify

                                                            1 √ 3

We can multiply both the numerator and denominator by √ 3
To rewrite the expression to have a rational denominator: 1 √ 3 = 1 √ 3 × √ 3 √ 3 = 1 × √ 3 √ 3 × √ 3 = √ 3 3 

We now have two different forms of the same number: 1 √ 3 = √ 3 3 

Rationalize the Denominator with Multiple Terms:

Suppose another but a little complex term in order to clarify the concepts 

4 * √64 / 3 * 3√27


First, we simplify 64 and 27 and after move forward.
4 * √4^2 * 4 / 3 * 3√3^2 * 3

4 * 8 / 3 * 3    


Working of Rationalize the Denominator Calculator:

In order to avoid square roots and rationalize the denominators use our best tool which is designed with user-friendliness in mind. 

What to Do?

  • First, select a simple or advanced method 
  • Select the expression from the drop-down menu 
  • Insert the values for the numerator and denominator
  • Tap on the “Calculate” icon

What to Get?

  • Rationalization of denominator 
  • Complete the calculation in steps 


Wikipedia: Rationalisation (mathematics), Rationalisation of a monomial square root and cube root, Dealing with more square roots.

Lumen Learning: Rationalize Denominators, Example. 

Alan Walker

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.

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