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Rationalize Denominator

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Last updated date: 13th Jul 2024
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How to Rationalize the Denominator?

Rationalizing the denominator means eliminating any radical expressions in the denominator, such as square roots and cube roots. The idea is to multiply the original fraction by a value so that after simplification, the denominator no longer contains any radical. 

What is a Rational Number?

A rational number, in Mathematics, can be defined as any number that can be represented in the form of p/q where q is valid for 0. Also, we can say that any fraction fits into the rational number category, where the denominator and the numerator are integers and the denominator is not equal to zero. If the rational number (i.e. fraction) is divided, the result will be in decimal form. 

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What is a Denominator?

The bottom number in a fraction of it. Shows how many equal parts of the item are divided into.

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What is Meant by Rationalizing?

In elementary algebra, rationalization is a method that eliminates radicals in the fraction denominator. Radicals are the inverse operation of exponents; we can undo a power with a radical, and so we can undo a radical with a power. For instance, if we square 9, we get 81, and if we take the square root of 81 we get 9.

Why Rationalize the Denominator?

The reason is that if we need to add or subtract radical fractions, it's easier to calculate if there are whole numbers in the denominator instead of irrational numbers. It's easier to add, for example, \[\frac{2\sqrt{3}}{3}\] + \[\frac{\sqrt{3}}{3}\] than \[\frac{2}{\sqrt{3}}\] + \[\frac{1}{2 + \sqrt{3}}\] 

To add the first set of fractions together, all we need to do is make a common denominator of 3 and then add the same terms from the numerators. It's not nearly as clear what the common denominator of the second set of fractions is. 

To solve the second problem, you would most likely rationalize the denominator first and then make the common denominator of \[\sqrt{3}\] and (2 + \[\sqrt{3}\]) before adding the fractions together.

How to Rationalize the Denominator with One Term?

Step 1: Multiply the numerator and the denominator by a radical to get rid of the radicals in the denominator.

Step 2: Make sure that all radicals are simplified.

Step 3: Simplify the fraction, if necessary.

For Example: Rationalize \[\frac{a}{\sqrt{b}}\] 


⇒  \[\frac{a}{\sqrt{b}}\] x \[\frac{\sqrt{b}}{\sqrt{b}}\]

⇒ \[\frac{a\sqrt{b}}{\sqrt{b} \ast \sqrt{b}}\]

⇒ \[\frac{a\sqrt{b}}{b}\] 

How to Rationalize the Denominator with Two Terms?

  • Step 1: To rationalize the denominator, both the numerator and the denominator must be multiplied by the conjugate of the denominator. Remember to find the conjugate all you need to do is to change the sign between the two terms.

  • Step 2: Distribute (or FOIL) the numerator and the denominator. Remember, you can multiply numbers outside the radical, numbers outside the radical, numbers inside the radical, numbers inside the radical.

  • Step 3: Like terms are combined in the next step.

  • Step 4: Standardize the radicals.

  • Step 5: Next step is to combine the like terms.

  • Step 6: If you can, reduce the fraction. To reduce the fraction, you must reduce the amount of EACH outside the radical by the same number. If you cannot reduce each number outside the radical by the same number, the fraction cannot be reduced.

For Example: Rationalize \[\frac{a}{c + \sqrt{b}}\] 


⇒ \[\frac{a}{c + \sqrt{b}}\] х \[\frac{c - \sqrt{b}}{c - \sqrt{b}}\]

⇒ \[\frac{a(c - \sqrt{b})}{c^{2} - b}\] 

⇒ \[\frac{ac - a\sqrt{b}}{c^{2} - b}\] 

How to Rationalize the Denominator Calculator?

Rationalize Denominator Calculator deals with four types of radical division: radical/radical, sum/radical, radical/sum, and sum/sum. Below, we focus on them one by one and describe how the rationalization of math works in each case.

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

The simplest case we could have. To rationalize the denominator and simplify the result, we need to multiply our expression by ᵏ√(yᵏ⁻¹) / ᵏ√(yᵏ⁻¹). This will be given in the denominator 

x * ᵏ√y * ᵏ√(yᵏ⁻¹) = x * ᵏ√(yᵏ) = x * y

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

Similar to point 1, but this time we've got two summands in the numerator. Luckily, this doesn't change the process much because we're only concerned with the math rationalization of the denominator, which happens to be the same thing. In other words, we're-multiply ᵏ√(yᵏ⁻¹) / ᵏ√(yᵏ⁻¹) and get x * y below as a result.

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

Here's when things get interesting. This time, we need to rationalize both of the denominator's summands, which means that we need to take care of both at the same time. To do this, we use the formula (a - b) * (a + b) = a² - b², which gives us two squares that kill square roots (note that this formula works only for radicals of order 2).

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

Similar to paragraph 3. Point 2 in the same way. It's been to point 1. Again, the extra summand above does not change the process: we still multiply the whole thing by (a * √b + c * √d) / (x * √y + z * √u). However, this time the multiplication of the radical expressions in the numerators gives four summaries: each element of the first time each of the seconds.

Solved Examples

1. Rationalize : 

a. \[\frac{1}{\sqrt{7}}\]


⇒ \[\frac{1}{\sqrt{7}}\] х \[\frac{\sqrt{7}}{\sqrt{7}}\]  




⇒ \[\frac{\sqrt{7}}{7}\]  

b. \[\frac{5}{3 \sqrt{5}}\]


⇒ \[\frac{5}{3 \sqrt{5}}\] х \[\frac{3\sqrt{5}}{3\sqrt{5}}\]

⇒ \[\frac{15 \sqrt{5}}{9 \times 5}\] ⇒ \[\frac{15 \sqrt{5}}{45}\]  


⇒ \[\frac{\sqrt{5}}{3}\] 

FAQs on Rationalize Denominator

1. How to Rationalize a Binomial Denominator?

Ans. Multiply the numerator and the denominator by the conjugate of the denominator to rationalize the radical expression. The conjugate of a binomial is obtained by changing the middle sign to the opposite.

2. What is the Rationalization of a Fraction?

Ans. "Rationalizing the denominator" is when we move the root (like a square root or a cube root) from the bottom of the fraction to the top.

3. How Do You Rationalize the Square of a Root?

Ans. In the denominator of the equation, you cannot have square roots. You need to multiply so that the square root is gone. You can do this by multiplying the top and bottom of the equation with the bottom denominator. From here, this is going to take the square root to go away, so your equation is going to be normal numbers.